research article spectral-collocation methods for...
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Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 821327 14 pageshttpdxdoiorg1011552013821327
Research ArticleSpectral-Collocation Methods for Fractional PantographDelay-Integrodifferential Equations
Yin Yang1 and Yunqing Huang2
1 College of Civil Engineering and Mechanics Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan University Xiangtan Hunan 411105 China
2Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan Hunan 411105 China
Correspondence should be addressed to Yin Yang yangyinxtuxtueducn
Received 15 May 2013 Accepted 15 September 2013
Academic Editor Varsha Daftardar-Gejji
Copyright copy 2013 Y Yang and Y Huang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We propose and analyze a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterratype with pantograph delay The fractional derivative is described in the Caputo sense We provide a rigorous error analysis for thecollocation method which shows that the error of approximate solution decays exponentially in 119871
infin norm and weighted 1198712-norm
The numerical examples are given to illustrate the theoretical results
1 Introduction
Many phenomena in engineering physics chemistry andother sciences can be described very successfully by modelsusing mathematical tools from fractional calculus that is thetheory of derivatives and integrals of fractional nonintegerorder This allows one to describe physical phenomenamore accurately Moreover fractional calculus is appliedto the model frequency dependent damping behavior ofmany viscoelastic materials economics and dynamics ofinterfaces between nanoparticles and substrates Recentlyseveral numerical methods to solve fractional differentialequations (FDEs) and fractional integrodifferential equations(FIDEs) have been proposed
In this paper we consider the general linear fractionalpantograph delay-integrodifferential equations (FDIDEs)with proportional delays
119863120574
119910 (119905) = 119910 (119905) + 119910 (119902119905) + 119892 (119905) + int
119905
0
1(119905 119904) 119910 (119904) 119889119904
+ int
119902119905
0
2(119905 120591) 119910 (120591) 119889120591
0 lt 120574 le 1 119905 isin 119868 = [0 119879]
119910 (0) = 1199100
(1)
with 0 lt 119902 lt 1 where 119892 119868 rarr 119877 1 119863 rarr 119877 (119863 =
(119905 120591) 0 le 119904 le 119905 le 119879) and 2 119863 rarr 119877 (119863 = (119905 120591)
0 le 120591 le 119902119905 119905 isin 119868) are given functions and are assumed tobe sufficiently smooth in the respective domains In (1) 119863120574
denotes the fractional derivative of fractional order 120574Differential and integral equations involving derivatives
of noninteger order have shown to be adequate modelsfor various phenomena arising in damping laws diffusionprocesses models of earthquake [1] fluid-dynamics trafficmodel [2] mathematical physics and engineering [3] fluidand continuum mechanics [4] chemistry acoustics andpsychology [5]
Let Γ(sdot) denote the Gamma function For any positiveinteger 119899 and 119899 minus 1 lt 120574 lt 119899 the Caputo derivative is definedas follows
119863120574
119910 (119905) =1
Γ (119899 minus 120574)int
119905
0
119910(119899)
(119904)
(119905 minus 119904)(120574minus119899+1)
119889119904 119905 isin [0 119879] (2)
The Riemann-Liouville fractional integral 119868120574 of order 120574 isdefined as
119868120574
119910 (119905) =1
Γ (120574)int
119905
0
(119905 minus 119904)120574minus1
119910 (119904) 119889119904 (3)
2 Advances in Mathematical Physics
we note that
119868120574
(119863120574
119910 (119905)) = 119910 (119905) minus
119899minus1
sum
119896=0
119910(119896)
(0)119905119896
119896 (4)
From (4) fractional integrodifferential equation (1) can bedescribed as
119863120574
119910 (119905) = 119910 (119905) + 119910 (119902119905) + 119892 (119905) + int
119905
0
1(119905 119904) 119910 (119904) 119889119904
+ int
119902119905
0
2(119905 120591) 119910 (120591) 119889120591
0 lt 120574 le 1 119905 isin 119868 = [0 119879]
119910 (119905) =1
Γ (120574)int
119905
0
(119905 minus 120591)120574minus1
119863120574
119910 (120591) 119889120591 + 119910 (0)
(5)
Several analytical methods have been introduced to solveFDEs including various transformation techniques [6] oper-ational calculus methods [7] the Adomian decompositionmethod [8] and the iterative and series-based methods [9]A small number of algorithms for the numerical solutionof FDEs have been suggested [10] and most of them arefinite difference methods which are generally limited to lowdimensions and are of limited accuracy
As we know fractional derivatives are global (they aredefined by an integral over the whole interval [0 119879]) andtherefore global methods such as spectral methods areperhaps better suited for FDEs Standard spectral methodspossess an infinite order of accuracy for the equations withregular solutions while failing for many complicated prob-lems with singular solutions So it is relevant to be interestedin how to enlarge the adaptability of spectral methods andconstruct certain simple approximation schemes without aloss of accuracy for more complicated problems
Spectral methods have been proposed to solve frac-tional differential equations such as the Legendre collocationmethod [11 12] Legendre wavelets method [13 14] andJacobi-Gauss-Lobatto collocation method [15] The authorsin [16ndash18] constructed an efficient spectral method for thenumerical approximation of fractional integrodifferentialequations based on tau and pseudospectral methods More-over Bhrawy et al [19] introduced a quadrature shiftedLegendre tau method based on the Gauss-Lobatto interpo-lation for solving multiorder FDEs with variable coefficientsand in [20] shifted Legendre spectral methods have beendeveloped for solving fractional-order multipoint boundaryvalue problems In [21] truncated Legendre series togetherwith the Legendre operationalmatrix of fractional derivativesare used for the numerical integration of fractional differ-ential equations In [22] the authors derived a new explicitformula for the integral of shifted Chebyshev polynomials ofany degree for any fractional-order The shifted Chebyshevoperational matrix [23] and shifted Jacobi operational matrix[24] of fractional derivatives have been developed which areapplied together with the spectral tau method for numericalsolution of general linear and nonlinear multiterm fractionaldifferential equations However very few theoretical results
were provided to justify the high accuracy numericallyobtained Recently Chen and Tang [25 26] developed anovel spectral Jacobi-collocation method to solve secondkindVolterra integral equationswith aweakly singular kerneland provided a rigorous error analysis which theoreticallyjustifies the spectral rate of convergence Inspired by thework of [26] we extend the approach to fractional orderdelay-integrodifferential equations (1) However it is difficultto apply the spectral approximations to the initial valueproblem and fractional order derivatives To facilitate the useof the spectral methods we restate the initial condition as anequivalent integral equation with singular kernel Then weget the discrete scheme by using Gauss quadrature formulaIn this paper we will provide a rigorous error analysis notonly for approximate solutions but also for approximatefractional derivatives which theoretically justifies the spectralrate of convergence
For ease of analysis we will describe the spectral methodson the standard interval 119868 = [minus1 1] Hence we employ thetransformation
119905 =119879
2(1 + 119909) 119904 =
119879 (1 + 120579)
2 120591 =
119879 (1 + 120578)
2
(6)
then the previous problem (5) becomes
119863120574
119906 (119909) = 119906 (119909) + 119906 (119902119909 + 119902 minus 1) + 119892 (119909)
+ int
119909
minus1
1198701(119909 120579) 119906 (120579) 119889120579
+ int
119902119909+119902minus1
minus1
1198702(119909 120578) 119906 (120578) 119889120578 119909 isin 119868
(7)
119906 (119909) =1
Γ (120574)(119879
2)
120574
int
119909
minus1
(119909 minus 119904)120574minus1
119863120574
119906 (119904) 119889119904 + 119906 (minus1)
(8)
where
119906 (119909) = 119910 (119879
2(1 + 119909)) 119863
120574
119906 (119909) = 119863120574
119910(119879
2(1 + 119909))
119892 (119909) =119879
2119892 (
119879
2(1 + 119909))
(9)
1198701(119909 120579) =
119879
41(119879
2(1 + 119909)
119879
2(1 + 120579))
1198702(119909 120578) =
119879
42(119879
2(1 + 119909)
119879
2(1 + 120578))
(10)
This paper is organized as follows In Section 2 we intro-duce the spectral approaches for pantograph FDIDEs Someuseful lemmas are provided in Section 3 These lemmas willplay a key role in the derivation of the convergence analysisWe provide a rigorous error analysis for the spectralmethodswhich shows that both the errors of approximate solutionsand the errors of approximate fractional derivatives of thesolutions decay exponentially in 119871
infin norm and weighted 1198712-
norm in Section 4 and Section 5 contains numerical results
Advances in Mathematical Physics 3
which will be used to verify the theoretical results obtained inSection 4
Throughout the paper 119862 will denote a generic positiveconstant that is independent of 119873 but which will depend on119902 119879 and on the bounds for the given functions 119886 119887 and 119870
119895
119895 = 1 2
2 Jacobi-Collocation Method
Let 120596120572120573(119909) = (1 minus 119909)120572
(1 + 119909)120573 be a weight function in the
usual sense for 120572 120573 gt minus1 The set of Jacobi polynomials119869120572120573
119899(119909)
infin
119899=0forms a complete1198712
120596120572120573(minus1 1)-orthogonal system
where 1198712120596120572120573(minus1 1) is a weighted space defined by
1198712
120596120572120573 (minus1 1) = V V is measurable and V
120596120572120573 lt infin
(11)
equipped with the norm
V120596120572120573 = (int
1
minus1
|V (119909)|2120596120572120573 (119909) 119889119909)12
(12)
and the inner product
(119906 V)120596120572120573 = int
1
minus1
119906 (119909) V (119909) 120596120572120573 (119909) 119889119909 forall119906 V isin 1198712
120596120572120573 (minus1 1)
(13)
For a given 119873 ge 0 we denote by 120579119896119873
119896=0the Legendre
points and by 120596119896119873
119896=0the corresponding Legendre weights
(ie Jacobi weights 12059600
119896119873
119896=0) Then the Legendre-Gauss
integration formula is
int
1
minus1
119891 (119909) 119889119909 asymp
119873
sum
119896=0
119891 (120579119896) 120596
119896 (14)
Similarly we denote by 120579119896119873
119896=0the Jacobi-Gauss points and by
120596120572120573
119896119873
119896=0the corresponding Jacobi weights Then the Jacobi-
Gauss integration formula is
int
1
minus1
119891 (119909) 120596120572120573
(119909) 119889119909 asymp
119873
sum
119896=0
119891 (120579119896) 120596
120572120573
119896 (15)
For a given positive integer119873 we denote the collocationpoints by 119909120572120573
119894119873
119894=0 which is the set of (119873 + 1) Jacobi-Gauss
points corresponding to the weight 120596120572120573(119909) Let P119873denote
the space of all polynomials of degree not exceeding 119873 Forany V isin 119862[minus1 1] we can define the Lagrange interpolatingpolynomial 119868120572120573
119873V isin P
119873 satisfying
119868120572120573
119873V (119909
119894) = V (119909
119894) 0 le 119894 le 119873 (16)
The Lagrange interpolating polynomial can be written in theform
119868120572120573
119873V (119909) =
119873
sum
119894minus0
V (119909119894) 119865
119894(119909) 0 le 119894 le 119873 (17)
where 119865119894(119909) is the Lagrange interpolation basis function
associated with 119909119894119873
119894=0
Let minus120583 = 120574 minus 1 In order to obtain high order accuracyof the approximate solution the main difficulty is to computethe integral terms in (7) and (8) In particular for small valuesof 119909 there is little information available for 119906 To overcomethis difficulty we transfer the integration interval to a fixedinterval [minus1 1] by using the following variable changes
120579 =1 + 119909
2] +
119909 minus 1
2≜ 120579 (119909 ])
120578 =119902 (1 + 119909)
2120585 +
119902 (1 + 119909)
2minus 1 ≜ 120578 (119909 120585)
(18)
119863120574
119906 (119909) = 119906 (119909) + 119906 (119902119909 + 119902 minus 1) + 119892 (119909)
+1 + 119909
2int
1
minus1
1198701(119909 120579 (119909 ])) 119906 (120579 (119909 ])) 119889]
+119902 (1 + 119909)
2int
1
minus1
1198702(119909 120578 (119909 120585)) 119906 (120578 (119909 120585)) 119889120585
(19)
119906 (119909) =1
Γ (120574)(1 + 119909
2)
120574
int
1
minus1
(1 minus ])minus120583119863120574
119906
times (120579 (119909 ])) 119889] + 119906 (minus1)
(20)
Set the collocation points as the set of (119873 + 1) Jacobi-Gauss points 119909minus120583minus120583
119894119873
119894=0associated with 120596
minus120583minus120583 Assume that(19) and (20) holds at 119909minus120583minus120583
119894
119863120574
119906 (119909minus120583minus120583
119894) = 119906 (119909
minus120583minus120583
119894) + 119906 (119902119909
minus120583minus120583
119894+ 119902 minus 1)
+ 119892 (119909minus120583minus120583
119894) +
1 + 119909minus120583minus120583
119894
2
times int
1
minus1
1198701(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
times 119906 (120579 (119909minus120583minus120583
119894 ])) 119889]
+
119902 (1 + 119909minus120583minus120583
119894)
2
times int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585))
times 119906 (120578 (119909minus120583minus120583
119894 120585)) 119889120585
(21)
119906 (119909minus120583minus120583
119894) =
1
Γ (120574)(
(1 + 119909minus120583minus120583
119894) 119879
4)
120574
times int
1
minus1
(1 minus ])minus120583119863120574
119906 (120579 (119909minus120583minus120583
119894 ])) 119889]
+ 119906 (minus1)
(22)
4 Advances in Mathematical Physics
Next using a (119873 + 1)-point Gauss quadrature formularelative to the Jacobi weight 12059600
119896119873
119894=0 the integration term in
(21) can be approximated by
int
1
minus1
1198701(119909
minus120583minus120583
120579 (119909minus120583minus120583
119894 ])) 119906 (120579 (119909minus120583minus120583
119894 ])) 119889]
=
119873
sum
119896=0
1198701(119909
119894 120579 (119909
119894 ]119896)) 119906 (120579 (119909
119894 ]119896)) 120596
00
119896
int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585)) 119906 (120578 (119909
minus120583minus120583
119894 120585)) 119889120585
=
119873
sum
119896=0
1198702(119909
119894 120578 (119909
119894 120585119896)) 119906 (120578 (119909
119894 120585119896)) 120596
00
119896
(23)
The sets ]119896119873
119896=0and 120585
119896119873
119896=0coincide with the Jacobi-Gauss
points corresponding Jacobi weights 12059600119896119873
119896=0 that is ]
119896119873
119896=0
and 120585119896119873
119896=0are Legendre-Gauss points
Using a (119873 + 1)-point Gauss quadrature formula relativeto the Jacobi weight 120596minus1205830 the integration term in (22) can beapproximated by
int
1
minus1
(1 minus ])120583119863120574
119906 (120579 (119909minus120583minus120583
119894 ])) 119889120579
=
119873
sum
119896=0
119863120574
119906 (120579 (119909minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896
(24)
where the set ]119896119873
119896=0is the Jacobi-Gauss points correspond-
ing to the weight 120596minus1205830We use 119906
119894 119906120574
119894to approximate the function value 119906(119909minus120583minus120583
119894)
119863120574
119906(119909minus120583minus120583
119894) 0 le 119894 le 119873 and expand 119906 and 119906120574 using Lagrange
interpolation polynomials that is
119880 (119909) =
119873
sum
119895=0
119906119895119865119895(119909) 119880
120574
(119909) =
119873
sum
119895=0
119906120574
119895119865119895(119909) (25)
where 119865119895(119909) is the Lagrange interpolation basis function
associated with 119909minus120583minus120583
119894119873
119894=0which is the set of (119873 + 1)
Jacobi-Gauss points The Jacobi collocation methods are toseek 119906
120574
119894119873
119894=0and 119906
119894119873
119894=0such that the following collocation
equations hold
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1))
+ 119892 (119909minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2
119873
sum
119895=0
119906119895(
119873
sum
119896=0
1198701(119909
minus120583minus120583
119894 120579 (119909
119894 ]119896))
times119865119895(120579 (119909
minus120583minus120583
119894 ]119896)) 120596
00
119896)
+
119902 (1 + 119909minus120583minus120583
119894)
2
119873
sum
119895=0
119906119895(
119873
sum
119896=0
1198702(119909
minus120583minus120583
119894 120578 (119909
119894 ]119896))
times 119865119895(120578 (119909
minus120583minus120583
119894 ]119896)) 120596
00
119896)
(26)
119906119894=
1
Γ (120574)(
(1 + 119909minus120583minus120583
119894) 119879
4)
120574
119873
sum
119895=0
119906120574
119895
times (
119873
sum
119896=0
119865119895(119904 (119909
minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896) + 119906 (minus1)
(27)
Writing 119880119873
= (1199060 119906
1 119906
119873)119879 and 119880
120574
119873= (119906
120574
0 119906
120574
1
119906120574
119873)119879 we obtain the following of the matrix form from (26)-
(27)
119880120574
119873= (119864 + 119871 + 119862 + 119863)119880
119873+ 119866
119880119873= 119880
minus1+ 119861119880
120574
119873
(28)
where
119864 is the identity matrix
119866 = (119892 (119909minus120583minus120583
0) 119892 (119909
minus120583minus120583
1) 119892 (119909
minus120583minus120583
119873))
119879
119871119894119895= 119865
119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)
119862119894119895=1 + 119909
minus120583minus120583
119894
2
119873
sum
119896=0
1(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]119896)) 119865
119895
times (120579 (119909119894 ]119896)) 120596
00
119896
119863119894119895=
119902 (1 + 119909minus120583minus120583
119894)
2
119873
sum
119896=0
2(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 ]119896)) 119865
119895
times (120578 (119909minus120583minus120583
119894 ]119896)) 120596
00
119896
119861119894119895=
1
Γ (120574)(
(1 + 119909minus120583minus120583
119894) 119879
4)
120574
times
119873
sum
119896=0
119865119895(119904 (119909
minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896
119880minus1
= (119906 (minus1) 119906 (minus1) 119906 (minus1))119879
(29)
We can get the values of 119906119894119873
119894=0and 119906
120574
119894119873
119894=0by solving the
system of linear system (28) Therefore the expressions of119880(119909) and 119880
120574
(119909) can be obtained
3 Some Useful Lemmas
In this section we will provide some elementary lemmaswhich are important for the derivation of the main results inthe subsequent section Let 119868 = (minus1 1)
Advances in Mathematical Physics 5
Lemma 1 (see [27]) Assume that an (119873 + 1)-point Gaussquadrature formula relative to the Jacobi weight is used tointegrate the product 119906120593 where 119906 isin 119867
119898
(119868)with 119868 for some119898 ge
1 and 120593 isin P119873 Then there exists a constant 119862 independent of
N such that100381610038161003816100381610038161003816100381610038161003816
int
1
minus1
119906 (119909) 120593 (119909) 119889119909 minus (119906 120593)119873
100381610038161003816100381610038161003816100381610038161003816
le 119862119873minus119898
|119906|119867119898119873
120596120572120573(119868)
100381710038171003817100381712059310038171003817100381710038171198712120596120572120573(119868)
(30)
where
|119906|119867119898119873
120596120572120573(119868)
= (
119898
sum
119895=min(119898119873+1)
10038171003817100381710038171003817119906(119895)10038171003817100381710038171003817
2
1198712
120596120572120573(119868)
)
12
(119906 120593)119873=
119873
sum
119895=0
119906 (119909119895) 120593 (119909
119895) 120596
119895
(31)
Lemma 2 (see [26 27]) Assume that 119906 isin 119867119898119873
120596minus120583minus120583
(119868) anddenote by 119868minus120583minus120583
119873119906 its interpolation polynomial associated with
the (119873 + 1) Jacobi-Gauss points 119909119895119873
119895=0 namely
119868minus120583minus120583
119873119906 =
119873
sum
119894=0
119906 (119909119894) 119865 (119909
119894) (32)
Then the following estimates hold
10038171003817100381710038171003817119906 minus 119868
minus120583minus120583
119873119906100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
|119906|119867119898119873
120596120572120573(119868) (33a)
10038171003817100381710038171003817119906 minus 119868
minus120583minus120583
11987311990610038171003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 le 120583 lt
1
2
11986211987312minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(33b)
where 120596119888 = 120596minus12minus12 denotes the Chebyshev weight function
Lemma 3 (see [28]) Assume that 119865119895(119909)
119873
119895=0are the 119873-th
degree Lagrange basis polynomials associated with the Gausspoints of the Jacobi polynomials Then
100381710038171003817100381710038171003817119868120572120573
119873
100381710038171003817100381710038171003817119871infin(119868)le max
119909isin[minus11]
119873
sum
119895=0
10038161003816100381610038161003816119865119895(119909)
10038161003816100381610038161003816
=
O (log119873) minus1 lt 120572 120573 le minus1
2
O (119873120574+12
) 120574 = max (120572 120573) 119900119905ℎ119890119903119908119894119904119890(34)
Lemma 4 (Gronwall inequality see [29] Lemma 711)Suppose that 119871 ge 0 0 lt 120583 lt 1 and 119906 and V are a nonnegativelocally integrable functions defined on [minus1 1] satisfying
119906 (119909) le V (119909) + 119871int
119909
minus1
(119909 minus 120591)minus120583
119906 (120591) 119889120591 (35)
Then there exists a constant 119862 = 119862(120583) such that
119906 (119909) le V (119909) + 119862119871int
119909
minus1
(119909 minus 120591)minus120583V (120591) 119889120591 119891119900119903 minus 1 le 119909 lt 1
(36)
Lemma 5 (see [30 31]) For a nonnegative integer 119903 and 120581 isin
(0 1) there exists a constant119862119903120581
gt 0 such that for any functionV isin 119862
119903120581
([minus1 1]) there exists a polynomial function T119873V isin
P119873such that
1003817100381710038171003817V minusT119873V1003817100381710038171003817119871infin(119868) le 119862
119903120581119873minus(119903+120581)
V119903120581 (37)
where sdot 119903120581
is the standard norm in 119862119903120581
([minus1 1]) which isdenoted by the space of functions whose 119903th derivatives areHolder continuous with exponent 120581 endowed with the usualnorm
V119903120581
= max0le120581le119903
max119909isin[minus11]
1003816100381610038161003816120597120581
119909V (119909)1003816100381610038161003816
+ max0le120581le119903
sup119909119910isin[minus11]119909 = 119910
1003816100381610038161003816120597120581
119909V (119909) minus 120597
120581
119909V (119909)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
120581
(38)
T119873is a linear operator from 119862
119903120581
([minus1 1]) intoP119873
Lemma 6 (see [32]) Let 120581 isin (0 1) and letM be defined by
(MV) (119909) = int
119909
minus1
(119909 minus 120591)minus120583
119870 (119909 120591) V (120591) 119889120591 (39)
Then for any function V isin 119862([minus1 1]) there exists a positiveconstant 119862 such that
10038161003816100381610038161003816MV (1199091015840) minusMV (11990910158401015840)
1003816100381610038161003816100381610038161003816100381610038161199091015840 minus 11990910158401015840
1003816100381610038161003816
le 119862 max119909isin[minus11]
|V (119909)| (40)
under the assumption that 0 lt 120581 lt 1 minus 120583 for any 1199091015840
11990910158401015840
isin
[minus1 1] and 1199091015840 = 11990910158401015840 This implies that
MV0120581
le 119862 max119909isin[minus11]
|V (119909)| 0 lt 120581 lt 1 minus 120583 (41)
Lemma7 (see [33]) For every bounded function V there existsa constant 119862 independent of V such that
sup119873
10038171003817100381710038171003817100381710038171003817100381710038171003817
119873
sum
119895=0
V (119909119895) 119865
119895(119909)
100381710038171003817100381710038171003817100381710038171003817100381710038171198712120596120572120573(119868)
le 119862 max119909isin[minus11]
|V (119909)| (42)
where 119865119895(119909) 119895 = 0 1 119873 are the Lagrange interpolation
basis functions associated with the Jacobi collocation points119909
119895119873
119895=0
Lemma 8 (see [34]) For all measurable function 119891 ge 0 thefollowing generalized Hardyrsquos inequality
(int
119887
119886
1003816100381610038161003816(119879119891) (119909)1003816100381610038161003816
119902
119906 (119909) 119889119909)
1119902
le (int
119887
119886
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901
V (119909) 119889119909)1119901
(43)
6 Advances in Mathematical Physics
holds if and only if
sup119886lt119909lt119887
(int
119887
119909
119906 (119905) 119889119905)
1119902
(int
119909
119886
V1minus1199011015840
(119905) 119889119905)
11199011015840
lt infin 1199011015840
=119901
119901 minus 1
(44)
for the case 1 lt 119901 le 119902 lt infin Here 119879 is an operator of the form
(119879119865) (119909) = int
119909
119886
119896 (119909 119905) 119891 (119905) 119889119905 (45)
with 119896(119909 119905) a given kernel 119906 V are nonnegative weightfunctions and minusinfin le 119886 lt 119887 le infin
4 Convergence Analysis
This section is devoted to provide a convergence analysis forthe numerical scheme The goal is to show that the rate ofconvergence is exponential that is the spectral accuracy canbe obtained for the proposed approximations Firstly we willcarry our convergence analysis in 119871
infin space
Theorem 9 Let 119906(119909) be the exact solution of the fractionaldelay-integrodifferential equation (7)-(8) which is assumedto be sufficiently smooth Assume that 119880(119909) and 119880
120574
(119909) areobtained by using the spectral collocation scheme (26)-(27)together with a polynomial interpolation (25) If 120574 associatedwith the weakly singular kernel satisfies 0 lt 120574 lt 1 and119906 isin 119867
119898+1
120596minus120583minus120583(119868) then
1003817100381710038171003817119880120574
minus 119863120574
1199061003817100381710038171003817119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(46)
119880 minus 119906119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(47)
provided that 119873 is sufficiently large where 119862 is a constantindependent of 119873 but which will depend on the bounds of thefunctions 119870(119909 119904) and the index 120583
119870lowast
= max119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
+ max119909isin[minus11]
10038161003816100381610038161198702(119909 120578 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(48)
U =1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868) (49)
Proof We let
(119870 (119904 (119909119894 ])) 120601 (119904 (119909
119894 ])))
119873
=
119873
sum
119896=0
119870(119904 (119909119894 ]119896)) 120601 (119904 (119909
119894 ]119896)) 120596
00
119896
(50)
The numerical scheme (26)-(27) can be written as
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2(119870
1(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
119880 (120579 (119909minus120583minus120583
119894 ])))
119873
+
119902 (1 + 119909minus120583minus120583
119894)
2(119870
2(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 ]))
119880 (120578 (119909minus120583minus120583
119894 ])))
119873
(51)
119906119894=
1
Γ (120574)(119879
2)
120574
int
119909119894
minus1
(119909119894minus 120579)
minus120583
119880120574
(120579) 119889120579 + 119906minus1 (52)
which gives
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2int
1
minus1
1198701(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
times 119880 (120579 (119909minus120583minus120583
119894 ])) 119889] + 119868
1(119909
minus120583minus120583
119894)
+
119902 (1 + 119909minus120583minus120583
119894)
2int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585))
times 119880 (120578 (119909minus120583minus120583
119894 120585)) 119889120585 + 119868
2(119909
minus120583minus120583
119894)
(53)
where
1198681(119909) =
1 + 119909
2int
1
minus1
1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]
minus1 + 119909
2(119870
1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))
119873
1198682(119909) =
119902 (1 + 119909)
2int
1
minus1
1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585
minus119902 (1 + 119909)
2(119870
2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))
119873
(54)
Advances in Mathematical Physics 7
Using the integration error estimates from Jacobi-Gausspolynomials quadrature in Lemma 1 we have
10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
119880 (120579 (119909 sdot))1198712(119868)
le 119862119873minus119898
119870lowast
1119880
1198712(119868)
(55)
10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)
le 119862119873minus119898
119870lowast
2119880
1198712(119868)
(56)
119870lowast
1= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
119870lowast
2= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(57)
From (18) (53) can be rewritten as
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579)119880 (120579) 119889120579 + 119868
1(119909
minus120583minus120583
119894)
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578)119880 (120578) 119889120578
+ 1198682(119909
minus120583minus120583
119894)
(58)
Let 119890 and 119890120574 denote the error function
119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574
(119909) = 119880120574
(119909) minus 119906120574
(119909)
(59)
we have
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119906 (120579) 119889120579
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119890 (120579) 119889120579
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119906 (120578) 119889120578
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119890 (120578) 119889120578
+ 1198681(119909
minus120583minus120583
119894) + 119868
2(119909
minus120583minus120583
119894)
(60)
Multiplying119865119894(119909)onboth sides of (62) and (52) summing
up from 0 to119873 and using (7)-(8) yield
119880120574
(119909) = 119880 (119909) + 119868minus120583minus120583
119873(
119873
sum
119895=0
119906119895119865119895(119902119909 + 119902 minus 1))
+ 119868minus120583minus120583
119873119892 (119909)
+ 119868minus120583minus120583
119873[119863
120574
119906 (119909) minus 119906 (119909)
minus119906 (119902 (119909) + 119902 minus 1) minus 119892 (119909)]
+ 119868minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ 119868minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909)
(61)
119880 (119909) = 119868minus120583minus120583
119873(119906 (119909) minus 119906
minus1)
+ 119868minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
+ 119868minus120583minus120583
119873(119906
minus1)
(62)
where
1198691(119909) = 119868
minus120583minus120583
1198731198681(119909) 119869
2(119909) = 119868
minus120583minus120583
1198731198682(119909) (63)
From (61)-(62) we have
119890120574
(119909) = 119890 (119909) + int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(64)
119890 (119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+ 1198694(119909) + 119869
8(119909)
(65)
where
1198693(119909) = 119868
minus120583minus120583
119873119863120574
119906 (119909) minus 119863120574
119906 (119909)
1198694(119909) = 119868
minus120583minus120583
119873119906 (119909) minus 119906 (119909)
1198695(119909) = 119868
minus120583minus120583
119873119906 (119902119909 + 119902 minus 1) minus 119906 (119902119909 + 119902 minus 1)
1198696(119909) = 119868
minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
minus int
119909
minus1
119870 (119909 119904) 119890 (119904) 119889119904
8 Advances in Mathematical Physics
1198697(119909) = 119868
minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
minus int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1198698(119909) = 119868
minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
minus1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
(66)
Due to (64) we obtain
119890120574
(119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
times 119890120574
(119911) 119889119911 119889120578 + 119867 (119909)
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(67)
where
119867(119909) = int
119909
minus1
1198701(119909 120579) int
120579
minus1
(1198694(120579) + 119869
8(120579)) 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(1198694(120578) + 119869
8(120578)) 119889120578
+ 1198694(119909) + 119869
8(119909)
(68)
Using the Dirichletrsquos formula which sates that
int
119909
minus1
int
120579
minus1
Φ (120579 120591) 119889120591 119889120579 = int
119909
minus1
int
119909
120591
Φ (120579 120591) 119889120579 119889120591 (69)
provided that the integral exists we obtain
1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
=1
Γ (120574)int
119909
minus1
(int
119909
120579
1198701(119909 120579) (120579 minus 120591)
minus120583
119889120579) 119890120574
(120591) 119889120591
le1
Γ (120574)21minus120583 max
119909120591isin[minus11]
100381610038161003816100381610038161(119909 120591)
10038161003816100381610038161003816int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
≜ 1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
(70)
letting 120578 = 119902120591 + 119902 minus 1 119911 = 119902120579 + 119902 minus 1 we have
1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
119890120574
(119911) 119889119911 119889120578
=1
Γ (120574)119902int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
119902120591+119902minus1
minus1
(119909 minus 119911)minus120583
119890120574
(119911) 119889119911) 119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
120591
minus1
(1 minus 120579)minus120583
119890120574
(119902120579 + 119902 minus 1) 119889120579)119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
(int
119909
120579
1198702(119909 119902120591 + 119902 minus 1)
times (1 minus 120579)
minus120583
119889120591) 119890120574
times (119902120579 + 119902 minus 1) 119889120579
le1
Γ (120574)21minus120583
1199021+120574 max
119909120591isin[minus11]
10038161003816100381610038161198702(119909 120591)
1003816100381610038161003816
times int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
≜ 1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
(71)
Then (67) gives
119890120574
(119909) le1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
+1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
+ 119867 (119909) + 1198691(119909) + 119869
2(119909) + 119869
3(119909)
+ 1198694(119909) + 119869
5(119909) + 119869
6(119909) + 119869
7(119909)
(72)
It follows from the Gronwall inequality that
1003817100381710038171003817119890120574
(119909)1003817100381710038171003817119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(73)
From (65) we have
119890 (119909)119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(74)
Advances in Mathematical Physics 9
We now apply Lemma 7 to obtain that
100381710038171003817100381711986911003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
100381710038171003817100381711986921003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
(75)
Due to Lemma 2
100381710038171003817100381711986931003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus1198981003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log1198731003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986941003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(76)
By virtue of Lemma 2 (33b) with119898 = 1
100381710038171003817100381711986961003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
100381710038171003817100381711986971003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
(77)
We now estimate the term 1198698(119909) It follows from Lemmas
5 and 6 with 119870(119909 120591) = 1Γ(120574) that
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
12057410038171003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873M119890
120574
)10038171003817100381710038171003817119871infin(119868)
le (1 +10038171003817100381710038171003817119868minus120583minus120583
119873
10038171003817100381710038171003817119871infin(119868))119862119873
minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581
le
119862119873(12)minus120583minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus120581 log1198731003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
1
2le 120583 lt 1
(78)
where in the last step we have used Lemma 6 under thefollowing assumption
1
2minus 120583 lt 120581 lt 1 minus 120583 when 0 lt 120583 lt
1
2
0 lt 120581 lt 1 minus 120583 when 1
2le 120583 lt 1
(79)
Provided that119873 is sufficiently large combining (75) (76)(77) and (78) gives1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)1003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
119880 (119909) minus 119906 (119909)119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
(80)
Using 120574 = 1 minus 120583 we have the desired estimates (46) and (47)
Next we will give the error estimates in 1198712
120596minus120583minus120583 space
Theorem 10 If the hypotheses given in Theorem 9 hold then1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)10038171003817100381710038171198712120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
119880(119909) minus 119906(119909)1198712
120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
(81)
10 Advances in Mathematical Physics
for any 120581 isin (0 120574) provided that119873 is sufficiently large and 119862 isa constant independent of119873 where
1198811= 119870
lowast
(1199061198712(119868)
+1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868))
1198812= 119870
lowast
1199061198712(119868)
(82)
Proof Byusing the generalization ofGronwalls Lemma 4 andthe Hardy inequality Lemma 8 it follows from (72) that
100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
1198901198712
120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
(83)
Now using Lemma 7 we have10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
1(119906
1198712(119868)
+ 119890119871infin(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
2(119906
1198712(119868)
+ 119890119871infin(119868))
(84)
By the convergence result in Theorem 9 (119898 = 1) we have
119890119871infin(119868)
le 119862(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868)) (85)
So that10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
1(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
2(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
(86)
Due to Lemma 2 (33a)10038171003817100381710038171198693
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1198981003816100381610038161003816119863120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
(87)
By virtue of Lemma 2 (33a) with119898 = 1
1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
10038161003816100381610038161003816100381610038161003816
int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
100381610038161003816100381610038161003816100381610038161003816
int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
(88)
Finally it follows from Lemmas 5 and 7 that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
120574100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le10038171003817100381710038171003817119868minus120583minus120583
119873(M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
+1003817100381710038171003817M119890
120574
minusT11987311989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)
le 1198621003817100381710038171003817M119890
120574
minusT1198731198901205741003817100381710038171003817119871infin(119868)
le 119862119873minus1205811003817100381710038171003817M119890
12057410038171003817100381710038170120581
le 119862119873minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
(89)
where in the last step we used Lemma 6 for any 120581 isin (0 1minus120583)By the convergence result in Theorem 9 we obtain that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)
le
11986211987312minus120583minus119898minus120581
(119870lowast
1199061198712(119868)
+ 11987312U) 0 lt 120583 lt
1
2
119862119873minus119898minus120581 log119873(119870
lowast
1199061198712(119868)
+ 11987312U)
1
2le 120583 lt 1
(90)
for119873 sufficiently large and for any 120581 isin (0 1 minus 120583) The desiredestimates (81) follows from the previous estimates and (83)with 120574 = 1 minus 120583
5 Algorithm Implementation andNumerical Experiments
In this subsection we present the numerical results obtainedby the proposed collocation spectral method The estimatesin Section 4 indicates that the convergence of numericalsolutions is exponential if the exact solution is smooth Toconfirm the theoretical prediction a numerical experimentis carried out by considering the following example
Example 1 Consider the following fractional integrodifferen-tial equation
119863075
119910 (119905) =6119905225
Γ (325)minus1199052
119890119905
5119910 (119905) minus
(119902119905)2
5119910 (119902119905)
+ int
119905
0
119890119905
119904119910 (119904) 119889119904 + int
119902119905
0
120591119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 0
(91)
The corresponding exact solution is given by 119910(119905) = 1199053
Figure 1 presents the approximate and exact solutionson the left-hand side and presents the approximate and
Advances in Mathematical Physics 11
0 02 04 06 08 10
02
04
06
08
1
12
14
Exact solutionApproximate solution
(a)
0 02 04 06 08 10
05
1
15
2
25
Exact solutionApproximate solution
(b)
Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075
119910(119905) (b)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(a)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(b)
Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075
119910(119905) (b) versusthe number of collocation points in 119871
infin and 1198712
120596norms
exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871
infin and 1198712
120596minus120583minus120583 norms As
expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions
Example 2 Consider the following fractional integrodiffer-ential equation
119863120572
119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int
119905
0
119890119904(119905minus119904)
119910 (119904) 119889119904
+ int
119902119905
0
119902119905119890120591(119902119905minus120591)
119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 1
(92)
when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Advances in Mathematical Physics
we note that
119868120574
(119863120574
119910 (119905)) = 119910 (119905) minus
119899minus1
sum
119896=0
119910(119896)
(0)119905119896
119896 (4)
From (4) fractional integrodifferential equation (1) can bedescribed as
119863120574
119910 (119905) = 119910 (119905) + 119910 (119902119905) + 119892 (119905) + int
119905
0
1(119905 119904) 119910 (119904) 119889119904
+ int
119902119905
0
2(119905 120591) 119910 (120591) 119889120591
0 lt 120574 le 1 119905 isin 119868 = [0 119879]
119910 (119905) =1
Γ (120574)int
119905
0
(119905 minus 120591)120574minus1
119863120574
119910 (120591) 119889120591 + 119910 (0)
(5)
Several analytical methods have been introduced to solveFDEs including various transformation techniques [6] oper-ational calculus methods [7] the Adomian decompositionmethod [8] and the iterative and series-based methods [9]A small number of algorithms for the numerical solutionof FDEs have been suggested [10] and most of them arefinite difference methods which are generally limited to lowdimensions and are of limited accuracy
As we know fractional derivatives are global (they aredefined by an integral over the whole interval [0 119879]) andtherefore global methods such as spectral methods areperhaps better suited for FDEs Standard spectral methodspossess an infinite order of accuracy for the equations withregular solutions while failing for many complicated prob-lems with singular solutions So it is relevant to be interestedin how to enlarge the adaptability of spectral methods andconstruct certain simple approximation schemes without aloss of accuracy for more complicated problems
Spectral methods have been proposed to solve frac-tional differential equations such as the Legendre collocationmethod [11 12] Legendre wavelets method [13 14] andJacobi-Gauss-Lobatto collocation method [15] The authorsin [16ndash18] constructed an efficient spectral method for thenumerical approximation of fractional integrodifferentialequations based on tau and pseudospectral methods More-over Bhrawy et al [19] introduced a quadrature shiftedLegendre tau method based on the Gauss-Lobatto interpo-lation for solving multiorder FDEs with variable coefficientsand in [20] shifted Legendre spectral methods have beendeveloped for solving fractional-order multipoint boundaryvalue problems In [21] truncated Legendre series togetherwith the Legendre operationalmatrix of fractional derivativesare used for the numerical integration of fractional differ-ential equations In [22] the authors derived a new explicitformula for the integral of shifted Chebyshev polynomials ofany degree for any fractional-order The shifted Chebyshevoperational matrix [23] and shifted Jacobi operational matrix[24] of fractional derivatives have been developed which areapplied together with the spectral tau method for numericalsolution of general linear and nonlinear multiterm fractionaldifferential equations However very few theoretical results
were provided to justify the high accuracy numericallyobtained Recently Chen and Tang [25 26] developed anovel spectral Jacobi-collocation method to solve secondkindVolterra integral equationswith aweakly singular kerneland provided a rigorous error analysis which theoreticallyjustifies the spectral rate of convergence Inspired by thework of [26] we extend the approach to fractional orderdelay-integrodifferential equations (1) However it is difficultto apply the spectral approximations to the initial valueproblem and fractional order derivatives To facilitate the useof the spectral methods we restate the initial condition as anequivalent integral equation with singular kernel Then weget the discrete scheme by using Gauss quadrature formulaIn this paper we will provide a rigorous error analysis notonly for approximate solutions but also for approximatefractional derivatives which theoretically justifies the spectralrate of convergence
For ease of analysis we will describe the spectral methodson the standard interval 119868 = [minus1 1] Hence we employ thetransformation
119905 =119879
2(1 + 119909) 119904 =
119879 (1 + 120579)
2 120591 =
119879 (1 + 120578)
2
(6)
then the previous problem (5) becomes
119863120574
119906 (119909) = 119906 (119909) + 119906 (119902119909 + 119902 minus 1) + 119892 (119909)
+ int
119909
minus1
1198701(119909 120579) 119906 (120579) 119889120579
+ int
119902119909+119902minus1
minus1
1198702(119909 120578) 119906 (120578) 119889120578 119909 isin 119868
(7)
119906 (119909) =1
Γ (120574)(119879
2)
120574
int
119909
minus1
(119909 minus 119904)120574minus1
119863120574
119906 (119904) 119889119904 + 119906 (minus1)
(8)
where
119906 (119909) = 119910 (119879
2(1 + 119909)) 119863
120574
119906 (119909) = 119863120574
119910(119879
2(1 + 119909))
119892 (119909) =119879
2119892 (
119879
2(1 + 119909))
(9)
1198701(119909 120579) =
119879
41(119879
2(1 + 119909)
119879
2(1 + 120579))
1198702(119909 120578) =
119879
42(119879
2(1 + 119909)
119879
2(1 + 120578))
(10)
This paper is organized as follows In Section 2 we intro-duce the spectral approaches for pantograph FDIDEs Someuseful lemmas are provided in Section 3 These lemmas willplay a key role in the derivation of the convergence analysisWe provide a rigorous error analysis for the spectralmethodswhich shows that both the errors of approximate solutionsand the errors of approximate fractional derivatives of thesolutions decay exponentially in 119871
infin norm and weighted 1198712-
norm in Section 4 and Section 5 contains numerical results
Advances in Mathematical Physics 3
which will be used to verify the theoretical results obtained inSection 4
Throughout the paper 119862 will denote a generic positiveconstant that is independent of 119873 but which will depend on119902 119879 and on the bounds for the given functions 119886 119887 and 119870
119895
119895 = 1 2
2 Jacobi-Collocation Method
Let 120596120572120573(119909) = (1 minus 119909)120572
(1 + 119909)120573 be a weight function in the
usual sense for 120572 120573 gt minus1 The set of Jacobi polynomials119869120572120573
119899(119909)
infin
119899=0forms a complete1198712
120596120572120573(minus1 1)-orthogonal system
where 1198712120596120572120573(minus1 1) is a weighted space defined by
1198712
120596120572120573 (minus1 1) = V V is measurable and V
120596120572120573 lt infin
(11)
equipped with the norm
V120596120572120573 = (int
1
minus1
|V (119909)|2120596120572120573 (119909) 119889119909)12
(12)
and the inner product
(119906 V)120596120572120573 = int
1
minus1
119906 (119909) V (119909) 120596120572120573 (119909) 119889119909 forall119906 V isin 1198712
120596120572120573 (minus1 1)
(13)
For a given 119873 ge 0 we denote by 120579119896119873
119896=0the Legendre
points and by 120596119896119873
119896=0the corresponding Legendre weights
(ie Jacobi weights 12059600
119896119873
119896=0) Then the Legendre-Gauss
integration formula is
int
1
minus1
119891 (119909) 119889119909 asymp
119873
sum
119896=0
119891 (120579119896) 120596
119896 (14)
Similarly we denote by 120579119896119873
119896=0the Jacobi-Gauss points and by
120596120572120573
119896119873
119896=0the corresponding Jacobi weights Then the Jacobi-
Gauss integration formula is
int
1
minus1
119891 (119909) 120596120572120573
(119909) 119889119909 asymp
119873
sum
119896=0
119891 (120579119896) 120596
120572120573
119896 (15)
For a given positive integer119873 we denote the collocationpoints by 119909120572120573
119894119873
119894=0 which is the set of (119873 + 1) Jacobi-Gauss
points corresponding to the weight 120596120572120573(119909) Let P119873denote
the space of all polynomials of degree not exceeding 119873 Forany V isin 119862[minus1 1] we can define the Lagrange interpolatingpolynomial 119868120572120573
119873V isin P
119873 satisfying
119868120572120573
119873V (119909
119894) = V (119909
119894) 0 le 119894 le 119873 (16)
The Lagrange interpolating polynomial can be written in theform
119868120572120573
119873V (119909) =
119873
sum
119894minus0
V (119909119894) 119865
119894(119909) 0 le 119894 le 119873 (17)
where 119865119894(119909) is the Lagrange interpolation basis function
associated with 119909119894119873
119894=0
Let minus120583 = 120574 minus 1 In order to obtain high order accuracyof the approximate solution the main difficulty is to computethe integral terms in (7) and (8) In particular for small valuesof 119909 there is little information available for 119906 To overcomethis difficulty we transfer the integration interval to a fixedinterval [minus1 1] by using the following variable changes
120579 =1 + 119909
2] +
119909 minus 1
2≜ 120579 (119909 ])
120578 =119902 (1 + 119909)
2120585 +
119902 (1 + 119909)
2minus 1 ≜ 120578 (119909 120585)
(18)
119863120574
119906 (119909) = 119906 (119909) + 119906 (119902119909 + 119902 minus 1) + 119892 (119909)
+1 + 119909
2int
1
minus1
1198701(119909 120579 (119909 ])) 119906 (120579 (119909 ])) 119889]
+119902 (1 + 119909)
2int
1
minus1
1198702(119909 120578 (119909 120585)) 119906 (120578 (119909 120585)) 119889120585
(19)
119906 (119909) =1
Γ (120574)(1 + 119909
2)
120574
int
1
minus1
(1 minus ])minus120583119863120574
119906
times (120579 (119909 ])) 119889] + 119906 (minus1)
(20)
Set the collocation points as the set of (119873 + 1) Jacobi-Gauss points 119909minus120583minus120583
119894119873
119894=0associated with 120596
minus120583minus120583 Assume that(19) and (20) holds at 119909minus120583minus120583
119894
119863120574
119906 (119909minus120583minus120583
119894) = 119906 (119909
minus120583minus120583
119894) + 119906 (119902119909
minus120583minus120583
119894+ 119902 minus 1)
+ 119892 (119909minus120583minus120583
119894) +
1 + 119909minus120583minus120583
119894
2
times int
1
minus1
1198701(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
times 119906 (120579 (119909minus120583minus120583
119894 ])) 119889]
+
119902 (1 + 119909minus120583minus120583
119894)
2
times int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585))
times 119906 (120578 (119909minus120583minus120583
119894 120585)) 119889120585
(21)
119906 (119909minus120583minus120583
119894) =
1
Γ (120574)(
(1 + 119909minus120583minus120583
119894) 119879
4)
120574
times int
1
minus1
(1 minus ])minus120583119863120574
119906 (120579 (119909minus120583minus120583
119894 ])) 119889]
+ 119906 (minus1)
(22)
4 Advances in Mathematical Physics
Next using a (119873 + 1)-point Gauss quadrature formularelative to the Jacobi weight 12059600
119896119873
119894=0 the integration term in
(21) can be approximated by
int
1
minus1
1198701(119909
minus120583minus120583
120579 (119909minus120583minus120583
119894 ])) 119906 (120579 (119909minus120583minus120583
119894 ])) 119889]
=
119873
sum
119896=0
1198701(119909
119894 120579 (119909
119894 ]119896)) 119906 (120579 (119909
119894 ]119896)) 120596
00
119896
int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585)) 119906 (120578 (119909
minus120583minus120583
119894 120585)) 119889120585
=
119873
sum
119896=0
1198702(119909
119894 120578 (119909
119894 120585119896)) 119906 (120578 (119909
119894 120585119896)) 120596
00
119896
(23)
The sets ]119896119873
119896=0and 120585
119896119873
119896=0coincide with the Jacobi-Gauss
points corresponding Jacobi weights 12059600119896119873
119896=0 that is ]
119896119873
119896=0
and 120585119896119873
119896=0are Legendre-Gauss points
Using a (119873 + 1)-point Gauss quadrature formula relativeto the Jacobi weight 120596minus1205830 the integration term in (22) can beapproximated by
int
1
minus1
(1 minus ])120583119863120574
119906 (120579 (119909minus120583minus120583
119894 ])) 119889120579
=
119873
sum
119896=0
119863120574
119906 (120579 (119909minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896
(24)
where the set ]119896119873
119896=0is the Jacobi-Gauss points correspond-
ing to the weight 120596minus1205830We use 119906
119894 119906120574
119894to approximate the function value 119906(119909minus120583minus120583
119894)
119863120574
119906(119909minus120583minus120583
119894) 0 le 119894 le 119873 and expand 119906 and 119906120574 using Lagrange
interpolation polynomials that is
119880 (119909) =
119873
sum
119895=0
119906119895119865119895(119909) 119880
120574
(119909) =
119873
sum
119895=0
119906120574
119895119865119895(119909) (25)
where 119865119895(119909) is the Lagrange interpolation basis function
associated with 119909minus120583minus120583
119894119873
119894=0which is the set of (119873 + 1)
Jacobi-Gauss points The Jacobi collocation methods are toseek 119906
120574
119894119873
119894=0and 119906
119894119873
119894=0such that the following collocation
equations hold
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1))
+ 119892 (119909minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2
119873
sum
119895=0
119906119895(
119873
sum
119896=0
1198701(119909
minus120583minus120583
119894 120579 (119909
119894 ]119896))
times119865119895(120579 (119909
minus120583minus120583
119894 ]119896)) 120596
00
119896)
+
119902 (1 + 119909minus120583minus120583
119894)
2
119873
sum
119895=0
119906119895(
119873
sum
119896=0
1198702(119909
minus120583minus120583
119894 120578 (119909
119894 ]119896))
times 119865119895(120578 (119909
minus120583minus120583
119894 ]119896)) 120596
00
119896)
(26)
119906119894=
1
Γ (120574)(
(1 + 119909minus120583minus120583
119894) 119879
4)
120574
119873
sum
119895=0
119906120574
119895
times (
119873
sum
119896=0
119865119895(119904 (119909
minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896) + 119906 (minus1)
(27)
Writing 119880119873
= (1199060 119906
1 119906
119873)119879 and 119880
120574
119873= (119906
120574
0 119906
120574
1
119906120574
119873)119879 we obtain the following of the matrix form from (26)-
(27)
119880120574
119873= (119864 + 119871 + 119862 + 119863)119880
119873+ 119866
119880119873= 119880
minus1+ 119861119880
120574
119873
(28)
where
119864 is the identity matrix
119866 = (119892 (119909minus120583minus120583
0) 119892 (119909
minus120583minus120583
1) 119892 (119909
minus120583minus120583
119873))
119879
119871119894119895= 119865
119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)
119862119894119895=1 + 119909
minus120583minus120583
119894
2
119873
sum
119896=0
1(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]119896)) 119865
119895
times (120579 (119909119894 ]119896)) 120596
00
119896
119863119894119895=
119902 (1 + 119909minus120583minus120583
119894)
2
119873
sum
119896=0
2(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 ]119896)) 119865
119895
times (120578 (119909minus120583minus120583
119894 ]119896)) 120596
00
119896
119861119894119895=
1
Γ (120574)(
(1 + 119909minus120583minus120583
119894) 119879
4)
120574
times
119873
sum
119896=0
119865119895(119904 (119909
minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896
119880minus1
= (119906 (minus1) 119906 (minus1) 119906 (minus1))119879
(29)
We can get the values of 119906119894119873
119894=0and 119906
120574
119894119873
119894=0by solving the
system of linear system (28) Therefore the expressions of119880(119909) and 119880
120574
(119909) can be obtained
3 Some Useful Lemmas
In this section we will provide some elementary lemmaswhich are important for the derivation of the main results inthe subsequent section Let 119868 = (minus1 1)
Advances in Mathematical Physics 5
Lemma 1 (see [27]) Assume that an (119873 + 1)-point Gaussquadrature formula relative to the Jacobi weight is used tointegrate the product 119906120593 where 119906 isin 119867
119898
(119868)with 119868 for some119898 ge
1 and 120593 isin P119873 Then there exists a constant 119862 independent of
N such that100381610038161003816100381610038161003816100381610038161003816
int
1
minus1
119906 (119909) 120593 (119909) 119889119909 minus (119906 120593)119873
100381610038161003816100381610038161003816100381610038161003816
le 119862119873minus119898
|119906|119867119898119873
120596120572120573(119868)
100381710038171003817100381712059310038171003817100381710038171198712120596120572120573(119868)
(30)
where
|119906|119867119898119873
120596120572120573(119868)
= (
119898
sum
119895=min(119898119873+1)
10038171003817100381710038171003817119906(119895)10038171003817100381710038171003817
2
1198712
120596120572120573(119868)
)
12
(119906 120593)119873=
119873
sum
119895=0
119906 (119909119895) 120593 (119909
119895) 120596
119895
(31)
Lemma 2 (see [26 27]) Assume that 119906 isin 119867119898119873
120596minus120583minus120583
(119868) anddenote by 119868minus120583minus120583
119873119906 its interpolation polynomial associated with
the (119873 + 1) Jacobi-Gauss points 119909119895119873
119895=0 namely
119868minus120583minus120583
119873119906 =
119873
sum
119894=0
119906 (119909119894) 119865 (119909
119894) (32)
Then the following estimates hold
10038171003817100381710038171003817119906 minus 119868
minus120583minus120583
119873119906100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
|119906|119867119898119873
120596120572120573(119868) (33a)
10038171003817100381710038171003817119906 minus 119868
minus120583minus120583
11987311990610038171003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 le 120583 lt
1
2
11986211987312minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(33b)
where 120596119888 = 120596minus12minus12 denotes the Chebyshev weight function
Lemma 3 (see [28]) Assume that 119865119895(119909)
119873
119895=0are the 119873-th
degree Lagrange basis polynomials associated with the Gausspoints of the Jacobi polynomials Then
100381710038171003817100381710038171003817119868120572120573
119873
100381710038171003817100381710038171003817119871infin(119868)le max
119909isin[minus11]
119873
sum
119895=0
10038161003816100381610038161003816119865119895(119909)
10038161003816100381610038161003816
=
O (log119873) minus1 lt 120572 120573 le minus1
2
O (119873120574+12
) 120574 = max (120572 120573) 119900119905ℎ119890119903119908119894119904119890(34)
Lemma 4 (Gronwall inequality see [29] Lemma 711)Suppose that 119871 ge 0 0 lt 120583 lt 1 and 119906 and V are a nonnegativelocally integrable functions defined on [minus1 1] satisfying
119906 (119909) le V (119909) + 119871int
119909
minus1
(119909 minus 120591)minus120583
119906 (120591) 119889120591 (35)
Then there exists a constant 119862 = 119862(120583) such that
119906 (119909) le V (119909) + 119862119871int
119909
minus1
(119909 minus 120591)minus120583V (120591) 119889120591 119891119900119903 minus 1 le 119909 lt 1
(36)
Lemma 5 (see [30 31]) For a nonnegative integer 119903 and 120581 isin
(0 1) there exists a constant119862119903120581
gt 0 such that for any functionV isin 119862
119903120581
([minus1 1]) there exists a polynomial function T119873V isin
P119873such that
1003817100381710038171003817V minusT119873V1003817100381710038171003817119871infin(119868) le 119862
119903120581119873minus(119903+120581)
V119903120581 (37)
where sdot 119903120581
is the standard norm in 119862119903120581
([minus1 1]) which isdenoted by the space of functions whose 119903th derivatives areHolder continuous with exponent 120581 endowed with the usualnorm
V119903120581
= max0le120581le119903
max119909isin[minus11]
1003816100381610038161003816120597120581
119909V (119909)1003816100381610038161003816
+ max0le120581le119903
sup119909119910isin[minus11]119909 = 119910
1003816100381610038161003816120597120581
119909V (119909) minus 120597
120581
119909V (119909)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
120581
(38)
T119873is a linear operator from 119862
119903120581
([minus1 1]) intoP119873
Lemma 6 (see [32]) Let 120581 isin (0 1) and letM be defined by
(MV) (119909) = int
119909
minus1
(119909 minus 120591)minus120583
119870 (119909 120591) V (120591) 119889120591 (39)
Then for any function V isin 119862([minus1 1]) there exists a positiveconstant 119862 such that
10038161003816100381610038161003816MV (1199091015840) minusMV (11990910158401015840)
1003816100381610038161003816100381610038161003816100381610038161199091015840 minus 11990910158401015840
1003816100381610038161003816
le 119862 max119909isin[minus11]
|V (119909)| (40)
under the assumption that 0 lt 120581 lt 1 minus 120583 for any 1199091015840
11990910158401015840
isin
[minus1 1] and 1199091015840 = 11990910158401015840 This implies that
MV0120581
le 119862 max119909isin[minus11]
|V (119909)| 0 lt 120581 lt 1 minus 120583 (41)
Lemma7 (see [33]) For every bounded function V there existsa constant 119862 independent of V such that
sup119873
10038171003817100381710038171003817100381710038171003817100381710038171003817
119873
sum
119895=0
V (119909119895) 119865
119895(119909)
100381710038171003817100381710038171003817100381710038171003817100381710038171198712120596120572120573(119868)
le 119862 max119909isin[minus11]
|V (119909)| (42)
where 119865119895(119909) 119895 = 0 1 119873 are the Lagrange interpolation
basis functions associated with the Jacobi collocation points119909
119895119873
119895=0
Lemma 8 (see [34]) For all measurable function 119891 ge 0 thefollowing generalized Hardyrsquos inequality
(int
119887
119886
1003816100381610038161003816(119879119891) (119909)1003816100381610038161003816
119902
119906 (119909) 119889119909)
1119902
le (int
119887
119886
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901
V (119909) 119889119909)1119901
(43)
6 Advances in Mathematical Physics
holds if and only if
sup119886lt119909lt119887
(int
119887
119909
119906 (119905) 119889119905)
1119902
(int
119909
119886
V1minus1199011015840
(119905) 119889119905)
11199011015840
lt infin 1199011015840
=119901
119901 minus 1
(44)
for the case 1 lt 119901 le 119902 lt infin Here 119879 is an operator of the form
(119879119865) (119909) = int
119909
119886
119896 (119909 119905) 119891 (119905) 119889119905 (45)
with 119896(119909 119905) a given kernel 119906 V are nonnegative weightfunctions and minusinfin le 119886 lt 119887 le infin
4 Convergence Analysis
This section is devoted to provide a convergence analysis forthe numerical scheme The goal is to show that the rate ofconvergence is exponential that is the spectral accuracy canbe obtained for the proposed approximations Firstly we willcarry our convergence analysis in 119871
infin space
Theorem 9 Let 119906(119909) be the exact solution of the fractionaldelay-integrodifferential equation (7)-(8) which is assumedto be sufficiently smooth Assume that 119880(119909) and 119880
120574
(119909) areobtained by using the spectral collocation scheme (26)-(27)together with a polynomial interpolation (25) If 120574 associatedwith the weakly singular kernel satisfies 0 lt 120574 lt 1 and119906 isin 119867
119898+1
120596minus120583minus120583(119868) then
1003817100381710038171003817119880120574
minus 119863120574
1199061003817100381710038171003817119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(46)
119880 minus 119906119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(47)
provided that 119873 is sufficiently large where 119862 is a constantindependent of 119873 but which will depend on the bounds of thefunctions 119870(119909 119904) and the index 120583
119870lowast
= max119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
+ max119909isin[minus11]
10038161003816100381610038161198702(119909 120578 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(48)
U =1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868) (49)
Proof We let
(119870 (119904 (119909119894 ])) 120601 (119904 (119909
119894 ])))
119873
=
119873
sum
119896=0
119870(119904 (119909119894 ]119896)) 120601 (119904 (119909
119894 ]119896)) 120596
00
119896
(50)
The numerical scheme (26)-(27) can be written as
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2(119870
1(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
119880 (120579 (119909minus120583minus120583
119894 ])))
119873
+
119902 (1 + 119909minus120583minus120583
119894)
2(119870
2(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 ]))
119880 (120578 (119909minus120583minus120583
119894 ])))
119873
(51)
119906119894=
1
Γ (120574)(119879
2)
120574
int
119909119894
minus1
(119909119894minus 120579)
minus120583
119880120574
(120579) 119889120579 + 119906minus1 (52)
which gives
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2int
1
minus1
1198701(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
times 119880 (120579 (119909minus120583minus120583
119894 ])) 119889] + 119868
1(119909
minus120583minus120583
119894)
+
119902 (1 + 119909minus120583minus120583
119894)
2int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585))
times 119880 (120578 (119909minus120583minus120583
119894 120585)) 119889120585 + 119868
2(119909
minus120583minus120583
119894)
(53)
where
1198681(119909) =
1 + 119909
2int
1
minus1
1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]
minus1 + 119909
2(119870
1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))
119873
1198682(119909) =
119902 (1 + 119909)
2int
1
minus1
1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585
minus119902 (1 + 119909)
2(119870
2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))
119873
(54)
Advances in Mathematical Physics 7
Using the integration error estimates from Jacobi-Gausspolynomials quadrature in Lemma 1 we have
10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
119880 (120579 (119909 sdot))1198712(119868)
le 119862119873minus119898
119870lowast
1119880
1198712(119868)
(55)
10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)
le 119862119873minus119898
119870lowast
2119880
1198712(119868)
(56)
119870lowast
1= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
119870lowast
2= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(57)
From (18) (53) can be rewritten as
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579)119880 (120579) 119889120579 + 119868
1(119909
minus120583minus120583
119894)
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578)119880 (120578) 119889120578
+ 1198682(119909
minus120583minus120583
119894)
(58)
Let 119890 and 119890120574 denote the error function
119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574
(119909) = 119880120574
(119909) minus 119906120574
(119909)
(59)
we have
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119906 (120579) 119889120579
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119890 (120579) 119889120579
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119906 (120578) 119889120578
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119890 (120578) 119889120578
+ 1198681(119909
minus120583minus120583
119894) + 119868
2(119909
minus120583minus120583
119894)
(60)
Multiplying119865119894(119909)onboth sides of (62) and (52) summing
up from 0 to119873 and using (7)-(8) yield
119880120574
(119909) = 119880 (119909) + 119868minus120583minus120583
119873(
119873
sum
119895=0
119906119895119865119895(119902119909 + 119902 minus 1))
+ 119868minus120583minus120583
119873119892 (119909)
+ 119868minus120583minus120583
119873[119863
120574
119906 (119909) minus 119906 (119909)
minus119906 (119902 (119909) + 119902 minus 1) minus 119892 (119909)]
+ 119868minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ 119868minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909)
(61)
119880 (119909) = 119868minus120583minus120583
119873(119906 (119909) minus 119906
minus1)
+ 119868minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
+ 119868minus120583minus120583
119873(119906
minus1)
(62)
where
1198691(119909) = 119868
minus120583minus120583
1198731198681(119909) 119869
2(119909) = 119868
minus120583minus120583
1198731198682(119909) (63)
From (61)-(62) we have
119890120574
(119909) = 119890 (119909) + int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(64)
119890 (119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+ 1198694(119909) + 119869
8(119909)
(65)
where
1198693(119909) = 119868
minus120583minus120583
119873119863120574
119906 (119909) minus 119863120574
119906 (119909)
1198694(119909) = 119868
minus120583minus120583
119873119906 (119909) minus 119906 (119909)
1198695(119909) = 119868
minus120583minus120583
119873119906 (119902119909 + 119902 minus 1) minus 119906 (119902119909 + 119902 minus 1)
1198696(119909) = 119868
minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
minus int
119909
minus1
119870 (119909 119904) 119890 (119904) 119889119904
8 Advances in Mathematical Physics
1198697(119909) = 119868
minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
minus int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1198698(119909) = 119868
minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
minus1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
(66)
Due to (64) we obtain
119890120574
(119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
times 119890120574
(119911) 119889119911 119889120578 + 119867 (119909)
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(67)
where
119867(119909) = int
119909
minus1
1198701(119909 120579) int
120579
minus1
(1198694(120579) + 119869
8(120579)) 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(1198694(120578) + 119869
8(120578)) 119889120578
+ 1198694(119909) + 119869
8(119909)
(68)
Using the Dirichletrsquos formula which sates that
int
119909
minus1
int
120579
minus1
Φ (120579 120591) 119889120591 119889120579 = int
119909
minus1
int
119909
120591
Φ (120579 120591) 119889120579 119889120591 (69)
provided that the integral exists we obtain
1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
=1
Γ (120574)int
119909
minus1
(int
119909
120579
1198701(119909 120579) (120579 minus 120591)
minus120583
119889120579) 119890120574
(120591) 119889120591
le1
Γ (120574)21minus120583 max
119909120591isin[minus11]
100381610038161003816100381610038161(119909 120591)
10038161003816100381610038161003816int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
≜ 1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
(70)
letting 120578 = 119902120591 + 119902 minus 1 119911 = 119902120579 + 119902 minus 1 we have
1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
119890120574
(119911) 119889119911 119889120578
=1
Γ (120574)119902int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
119902120591+119902minus1
minus1
(119909 minus 119911)minus120583
119890120574
(119911) 119889119911) 119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
120591
minus1
(1 minus 120579)minus120583
119890120574
(119902120579 + 119902 minus 1) 119889120579)119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
(int
119909
120579
1198702(119909 119902120591 + 119902 minus 1)
times (1 minus 120579)
minus120583
119889120591) 119890120574
times (119902120579 + 119902 minus 1) 119889120579
le1
Γ (120574)21minus120583
1199021+120574 max
119909120591isin[minus11]
10038161003816100381610038161198702(119909 120591)
1003816100381610038161003816
times int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
≜ 1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
(71)
Then (67) gives
119890120574
(119909) le1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
+1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
+ 119867 (119909) + 1198691(119909) + 119869
2(119909) + 119869
3(119909)
+ 1198694(119909) + 119869
5(119909) + 119869
6(119909) + 119869
7(119909)
(72)
It follows from the Gronwall inequality that
1003817100381710038171003817119890120574
(119909)1003817100381710038171003817119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(73)
From (65) we have
119890 (119909)119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(74)
Advances in Mathematical Physics 9
We now apply Lemma 7 to obtain that
100381710038171003817100381711986911003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
100381710038171003817100381711986921003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
(75)
Due to Lemma 2
100381710038171003817100381711986931003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus1198981003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log1198731003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986941003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(76)
By virtue of Lemma 2 (33b) with119898 = 1
100381710038171003817100381711986961003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
100381710038171003817100381711986971003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
(77)
We now estimate the term 1198698(119909) It follows from Lemmas
5 and 6 with 119870(119909 120591) = 1Γ(120574) that
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
12057410038171003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873M119890
120574
)10038171003817100381710038171003817119871infin(119868)
le (1 +10038171003817100381710038171003817119868minus120583minus120583
119873
10038171003817100381710038171003817119871infin(119868))119862119873
minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581
le
119862119873(12)minus120583minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus120581 log1198731003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
1
2le 120583 lt 1
(78)
where in the last step we have used Lemma 6 under thefollowing assumption
1
2minus 120583 lt 120581 lt 1 minus 120583 when 0 lt 120583 lt
1
2
0 lt 120581 lt 1 minus 120583 when 1
2le 120583 lt 1
(79)
Provided that119873 is sufficiently large combining (75) (76)(77) and (78) gives1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)1003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
119880 (119909) minus 119906 (119909)119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
(80)
Using 120574 = 1 minus 120583 we have the desired estimates (46) and (47)
Next we will give the error estimates in 1198712
120596minus120583minus120583 space
Theorem 10 If the hypotheses given in Theorem 9 hold then1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)10038171003817100381710038171198712120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
119880(119909) minus 119906(119909)1198712
120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
(81)
10 Advances in Mathematical Physics
for any 120581 isin (0 120574) provided that119873 is sufficiently large and 119862 isa constant independent of119873 where
1198811= 119870
lowast
(1199061198712(119868)
+1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868))
1198812= 119870
lowast
1199061198712(119868)
(82)
Proof Byusing the generalization ofGronwalls Lemma 4 andthe Hardy inequality Lemma 8 it follows from (72) that
100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
1198901198712
120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
(83)
Now using Lemma 7 we have10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
1(119906
1198712(119868)
+ 119890119871infin(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
2(119906
1198712(119868)
+ 119890119871infin(119868))
(84)
By the convergence result in Theorem 9 (119898 = 1) we have
119890119871infin(119868)
le 119862(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868)) (85)
So that10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
1(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
2(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
(86)
Due to Lemma 2 (33a)10038171003817100381710038171198693
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1198981003816100381610038161003816119863120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
(87)
By virtue of Lemma 2 (33a) with119898 = 1
1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
10038161003816100381610038161003816100381610038161003816
int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
100381610038161003816100381610038161003816100381610038161003816
int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
(88)
Finally it follows from Lemmas 5 and 7 that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
120574100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le10038171003817100381710038171003817119868minus120583minus120583
119873(M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
+1003817100381710038171003817M119890
120574
minusT11987311989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)
le 1198621003817100381710038171003817M119890
120574
minusT1198731198901205741003817100381710038171003817119871infin(119868)
le 119862119873minus1205811003817100381710038171003817M119890
12057410038171003817100381710038170120581
le 119862119873minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
(89)
where in the last step we used Lemma 6 for any 120581 isin (0 1minus120583)By the convergence result in Theorem 9 we obtain that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)
le
11986211987312minus120583minus119898minus120581
(119870lowast
1199061198712(119868)
+ 11987312U) 0 lt 120583 lt
1
2
119862119873minus119898minus120581 log119873(119870
lowast
1199061198712(119868)
+ 11987312U)
1
2le 120583 lt 1
(90)
for119873 sufficiently large and for any 120581 isin (0 1 minus 120583) The desiredestimates (81) follows from the previous estimates and (83)with 120574 = 1 minus 120583
5 Algorithm Implementation andNumerical Experiments
In this subsection we present the numerical results obtainedby the proposed collocation spectral method The estimatesin Section 4 indicates that the convergence of numericalsolutions is exponential if the exact solution is smooth Toconfirm the theoretical prediction a numerical experimentis carried out by considering the following example
Example 1 Consider the following fractional integrodifferen-tial equation
119863075
119910 (119905) =6119905225
Γ (325)minus1199052
119890119905
5119910 (119905) minus
(119902119905)2
5119910 (119902119905)
+ int
119905
0
119890119905
119904119910 (119904) 119889119904 + int
119902119905
0
120591119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 0
(91)
The corresponding exact solution is given by 119910(119905) = 1199053
Figure 1 presents the approximate and exact solutionson the left-hand side and presents the approximate and
Advances in Mathematical Physics 11
0 02 04 06 08 10
02
04
06
08
1
12
14
Exact solutionApproximate solution
(a)
0 02 04 06 08 10
05
1
15
2
25
Exact solutionApproximate solution
(b)
Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075
119910(119905) (b)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(a)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(b)
Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075
119910(119905) (b) versusthe number of collocation points in 119871
infin and 1198712
120596norms
exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871
infin and 1198712
120596minus120583minus120583 norms As
expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions
Example 2 Consider the following fractional integrodiffer-ential equation
119863120572
119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int
119905
0
119890119904(119905minus119904)
119910 (119904) 119889119904
+ int
119902119905
0
119902119905119890120591(119902119905minus120591)
119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 1
(92)
when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 3
which will be used to verify the theoretical results obtained inSection 4
Throughout the paper 119862 will denote a generic positiveconstant that is independent of 119873 but which will depend on119902 119879 and on the bounds for the given functions 119886 119887 and 119870
119895
119895 = 1 2
2 Jacobi-Collocation Method
Let 120596120572120573(119909) = (1 minus 119909)120572
(1 + 119909)120573 be a weight function in the
usual sense for 120572 120573 gt minus1 The set of Jacobi polynomials119869120572120573
119899(119909)
infin
119899=0forms a complete1198712
120596120572120573(minus1 1)-orthogonal system
where 1198712120596120572120573(minus1 1) is a weighted space defined by
1198712
120596120572120573 (minus1 1) = V V is measurable and V
120596120572120573 lt infin
(11)
equipped with the norm
V120596120572120573 = (int
1
minus1
|V (119909)|2120596120572120573 (119909) 119889119909)12
(12)
and the inner product
(119906 V)120596120572120573 = int
1
minus1
119906 (119909) V (119909) 120596120572120573 (119909) 119889119909 forall119906 V isin 1198712
120596120572120573 (minus1 1)
(13)
For a given 119873 ge 0 we denote by 120579119896119873
119896=0the Legendre
points and by 120596119896119873
119896=0the corresponding Legendre weights
(ie Jacobi weights 12059600
119896119873
119896=0) Then the Legendre-Gauss
integration formula is
int
1
minus1
119891 (119909) 119889119909 asymp
119873
sum
119896=0
119891 (120579119896) 120596
119896 (14)
Similarly we denote by 120579119896119873
119896=0the Jacobi-Gauss points and by
120596120572120573
119896119873
119896=0the corresponding Jacobi weights Then the Jacobi-
Gauss integration formula is
int
1
minus1
119891 (119909) 120596120572120573
(119909) 119889119909 asymp
119873
sum
119896=0
119891 (120579119896) 120596
120572120573
119896 (15)
For a given positive integer119873 we denote the collocationpoints by 119909120572120573
119894119873
119894=0 which is the set of (119873 + 1) Jacobi-Gauss
points corresponding to the weight 120596120572120573(119909) Let P119873denote
the space of all polynomials of degree not exceeding 119873 Forany V isin 119862[minus1 1] we can define the Lagrange interpolatingpolynomial 119868120572120573
119873V isin P
119873 satisfying
119868120572120573
119873V (119909
119894) = V (119909
119894) 0 le 119894 le 119873 (16)
The Lagrange interpolating polynomial can be written in theform
119868120572120573
119873V (119909) =
119873
sum
119894minus0
V (119909119894) 119865
119894(119909) 0 le 119894 le 119873 (17)
where 119865119894(119909) is the Lagrange interpolation basis function
associated with 119909119894119873
119894=0
Let minus120583 = 120574 minus 1 In order to obtain high order accuracyof the approximate solution the main difficulty is to computethe integral terms in (7) and (8) In particular for small valuesof 119909 there is little information available for 119906 To overcomethis difficulty we transfer the integration interval to a fixedinterval [minus1 1] by using the following variable changes
120579 =1 + 119909
2] +
119909 minus 1
2≜ 120579 (119909 ])
120578 =119902 (1 + 119909)
2120585 +
119902 (1 + 119909)
2minus 1 ≜ 120578 (119909 120585)
(18)
119863120574
119906 (119909) = 119906 (119909) + 119906 (119902119909 + 119902 minus 1) + 119892 (119909)
+1 + 119909
2int
1
minus1
1198701(119909 120579 (119909 ])) 119906 (120579 (119909 ])) 119889]
+119902 (1 + 119909)
2int
1
minus1
1198702(119909 120578 (119909 120585)) 119906 (120578 (119909 120585)) 119889120585
(19)
119906 (119909) =1
Γ (120574)(1 + 119909
2)
120574
int
1
minus1
(1 minus ])minus120583119863120574
119906
times (120579 (119909 ])) 119889] + 119906 (minus1)
(20)
Set the collocation points as the set of (119873 + 1) Jacobi-Gauss points 119909minus120583minus120583
119894119873
119894=0associated with 120596
minus120583minus120583 Assume that(19) and (20) holds at 119909minus120583minus120583
119894
119863120574
119906 (119909minus120583minus120583
119894) = 119906 (119909
minus120583minus120583
119894) + 119906 (119902119909
minus120583minus120583
119894+ 119902 minus 1)
+ 119892 (119909minus120583minus120583
119894) +
1 + 119909minus120583minus120583
119894
2
times int
1
minus1
1198701(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
times 119906 (120579 (119909minus120583minus120583
119894 ])) 119889]
+
119902 (1 + 119909minus120583minus120583
119894)
2
times int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585))
times 119906 (120578 (119909minus120583minus120583
119894 120585)) 119889120585
(21)
119906 (119909minus120583minus120583
119894) =
1
Γ (120574)(
(1 + 119909minus120583minus120583
119894) 119879
4)
120574
times int
1
minus1
(1 minus ])minus120583119863120574
119906 (120579 (119909minus120583minus120583
119894 ])) 119889]
+ 119906 (minus1)
(22)
4 Advances in Mathematical Physics
Next using a (119873 + 1)-point Gauss quadrature formularelative to the Jacobi weight 12059600
119896119873
119894=0 the integration term in
(21) can be approximated by
int
1
minus1
1198701(119909
minus120583minus120583
120579 (119909minus120583minus120583
119894 ])) 119906 (120579 (119909minus120583minus120583
119894 ])) 119889]
=
119873
sum
119896=0
1198701(119909
119894 120579 (119909
119894 ]119896)) 119906 (120579 (119909
119894 ]119896)) 120596
00
119896
int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585)) 119906 (120578 (119909
minus120583minus120583
119894 120585)) 119889120585
=
119873
sum
119896=0
1198702(119909
119894 120578 (119909
119894 120585119896)) 119906 (120578 (119909
119894 120585119896)) 120596
00
119896
(23)
The sets ]119896119873
119896=0and 120585
119896119873
119896=0coincide with the Jacobi-Gauss
points corresponding Jacobi weights 12059600119896119873
119896=0 that is ]
119896119873
119896=0
and 120585119896119873
119896=0are Legendre-Gauss points
Using a (119873 + 1)-point Gauss quadrature formula relativeto the Jacobi weight 120596minus1205830 the integration term in (22) can beapproximated by
int
1
minus1
(1 minus ])120583119863120574
119906 (120579 (119909minus120583minus120583
119894 ])) 119889120579
=
119873
sum
119896=0
119863120574
119906 (120579 (119909minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896
(24)
where the set ]119896119873
119896=0is the Jacobi-Gauss points correspond-
ing to the weight 120596minus1205830We use 119906
119894 119906120574
119894to approximate the function value 119906(119909minus120583minus120583
119894)
119863120574
119906(119909minus120583minus120583
119894) 0 le 119894 le 119873 and expand 119906 and 119906120574 using Lagrange
interpolation polynomials that is
119880 (119909) =
119873
sum
119895=0
119906119895119865119895(119909) 119880
120574
(119909) =
119873
sum
119895=0
119906120574
119895119865119895(119909) (25)
where 119865119895(119909) is the Lagrange interpolation basis function
associated with 119909minus120583minus120583
119894119873
119894=0which is the set of (119873 + 1)
Jacobi-Gauss points The Jacobi collocation methods are toseek 119906
120574
119894119873
119894=0and 119906
119894119873
119894=0such that the following collocation
equations hold
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1))
+ 119892 (119909minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2
119873
sum
119895=0
119906119895(
119873
sum
119896=0
1198701(119909
minus120583minus120583
119894 120579 (119909
119894 ]119896))
times119865119895(120579 (119909
minus120583minus120583
119894 ]119896)) 120596
00
119896)
+
119902 (1 + 119909minus120583minus120583
119894)
2
119873
sum
119895=0
119906119895(
119873
sum
119896=0
1198702(119909
minus120583minus120583
119894 120578 (119909
119894 ]119896))
times 119865119895(120578 (119909
minus120583minus120583
119894 ]119896)) 120596
00
119896)
(26)
119906119894=
1
Γ (120574)(
(1 + 119909minus120583minus120583
119894) 119879
4)
120574
119873
sum
119895=0
119906120574
119895
times (
119873
sum
119896=0
119865119895(119904 (119909
minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896) + 119906 (minus1)
(27)
Writing 119880119873
= (1199060 119906
1 119906
119873)119879 and 119880
120574
119873= (119906
120574
0 119906
120574
1
119906120574
119873)119879 we obtain the following of the matrix form from (26)-
(27)
119880120574
119873= (119864 + 119871 + 119862 + 119863)119880
119873+ 119866
119880119873= 119880
minus1+ 119861119880
120574
119873
(28)
where
119864 is the identity matrix
119866 = (119892 (119909minus120583minus120583
0) 119892 (119909
minus120583minus120583
1) 119892 (119909
minus120583minus120583
119873))
119879
119871119894119895= 119865
119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)
119862119894119895=1 + 119909
minus120583minus120583
119894
2
119873
sum
119896=0
1(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]119896)) 119865
119895
times (120579 (119909119894 ]119896)) 120596
00
119896
119863119894119895=
119902 (1 + 119909minus120583minus120583
119894)
2
119873
sum
119896=0
2(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 ]119896)) 119865
119895
times (120578 (119909minus120583minus120583
119894 ]119896)) 120596
00
119896
119861119894119895=
1
Γ (120574)(
(1 + 119909minus120583minus120583
119894) 119879
4)
120574
times
119873
sum
119896=0
119865119895(119904 (119909
minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896
119880minus1
= (119906 (minus1) 119906 (minus1) 119906 (minus1))119879
(29)
We can get the values of 119906119894119873
119894=0and 119906
120574
119894119873
119894=0by solving the
system of linear system (28) Therefore the expressions of119880(119909) and 119880
120574
(119909) can be obtained
3 Some Useful Lemmas
In this section we will provide some elementary lemmaswhich are important for the derivation of the main results inthe subsequent section Let 119868 = (minus1 1)
Advances in Mathematical Physics 5
Lemma 1 (see [27]) Assume that an (119873 + 1)-point Gaussquadrature formula relative to the Jacobi weight is used tointegrate the product 119906120593 where 119906 isin 119867
119898
(119868)with 119868 for some119898 ge
1 and 120593 isin P119873 Then there exists a constant 119862 independent of
N such that100381610038161003816100381610038161003816100381610038161003816
int
1
minus1
119906 (119909) 120593 (119909) 119889119909 minus (119906 120593)119873
100381610038161003816100381610038161003816100381610038161003816
le 119862119873minus119898
|119906|119867119898119873
120596120572120573(119868)
100381710038171003817100381712059310038171003817100381710038171198712120596120572120573(119868)
(30)
where
|119906|119867119898119873
120596120572120573(119868)
= (
119898
sum
119895=min(119898119873+1)
10038171003817100381710038171003817119906(119895)10038171003817100381710038171003817
2
1198712
120596120572120573(119868)
)
12
(119906 120593)119873=
119873
sum
119895=0
119906 (119909119895) 120593 (119909
119895) 120596
119895
(31)
Lemma 2 (see [26 27]) Assume that 119906 isin 119867119898119873
120596minus120583minus120583
(119868) anddenote by 119868minus120583minus120583
119873119906 its interpolation polynomial associated with
the (119873 + 1) Jacobi-Gauss points 119909119895119873
119895=0 namely
119868minus120583minus120583
119873119906 =
119873
sum
119894=0
119906 (119909119894) 119865 (119909
119894) (32)
Then the following estimates hold
10038171003817100381710038171003817119906 minus 119868
minus120583minus120583
119873119906100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
|119906|119867119898119873
120596120572120573(119868) (33a)
10038171003817100381710038171003817119906 minus 119868
minus120583minus120583
11987311990610038171003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 le 120583 lt
1
2
11986211987312minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(33b)
where 120596119888 = 120596minus12minus12 denotes the Chebyshev weight function
Lemma 3 (see [28]) Assume that 119865119895(119909)
119873
119895=0are the 119873-th
degree Lagrange basis polynomials associated with the Gausspoints of the Jacobi polynomials Then
100381710038171003817100381710038171003817119868120572120573
119873
100381710038171003817100381710038171003817119871infin(119868)le max
119909isin[minus11]
119873
sum
119895=0
10038161003816100381610038161003816119865119895(119909)
10038161003816100381610038161003816
=
O (log119873) minus1 lt 120572 120573 le minus1
2
O (119873120574+12
) 120574 = max (120572 120573) 119900119905ℎ119890119903119908119894119904119890(34)
Lemma 4 (Gronwall inequality see [29] Lemma 711)Suppose that 119871 ge 0 0 lt 120583 lt 1 and 119906 and V are a nonnegativelocally integrable functions defined on [minus1 1] satisfying
119906 (119909) le V (119909) + 119871int
119909
minus1
(119909 minus 120591)minus120583
119906 (120591) 119889120591 (35)
Then there exists a constant 119862 = 119862(120583) such that
119906 (119909) le V (119909) + 119862119871int
119909
minus1
(119909 minus 120591)minus120583V (120591) 119889120591 119891119900119903 minus 1 le 119909 lt 1
(36)
Lemma 5 (see [30 31]) For a nonnegative integer 119903 and 120581 isin
(0 1) there exists a constant119862119903120581
gt 0 such that for any functionV isin 119862
119903120581
([minus1 1]) there exists a polynomial function T119873V isin
P119873such that
1003817100381710038171003817V minusT119873V1003817100381710038171003817119871infin(119868) le 119862
119903120581119873minus(119903+120581)
V119903120581 (37)
where sdot 119903120581
is the standard norm in 119862119903120581
([minus1 1]) which isdenoted by the space of functions whose 119903th derivatives areHolder continuous with exponent 120581 endowed with the usualnorm
V119903120581
= max0le120581le119903
max119909isin[minus11]
1003816100381610038161003816120597120581
119909V (119909)1003816100381610038161003816
+ max0le120581le119903
sup119909119910isin[minus11]119909 = 119910
1003816100381610038161003816120597120581
119909V (119909) minus 120597
120581
119909V (119909)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
120581
(38)
T119873is a linear operator from 119862
119903120581
([minus1 1]) intoP119873
Lemma 6 (see [32]) Let 120581 isin (0 1) and letM be defined by
(MV) (119909) = int
119909
minus1
(119909 minus 120591)minus120583
119870 (119909 120591) V (120591) 119889120591 (39)
Then for any function V isin 119862([minus1 1]) there exists a positiveconstant 119862 such that
10038161003816100381610038161003816MV (1199091015840) minusMV (11990910158401015840)
1003816100381610038161003816100381610038161003816100381610038161199091015840 minus 11990910158401015840
1003816100381610038161003816
le 119862 max119909isin[minus11]
|V (119909)| (40)
under the assumption that 0 lt 120581 lt 1 minus 120583 for any 1199091015840
11990910158401015840
isin
[minus1 1] and 1199091015840 = 11990910158401015840 This implies that
MV0120581
le 119862 max119909isin[minus11]
|V (119909)| 0 lt 120581 lt 1 minus 120583 (41)
Lemma7 (see [33]) For every bounded function V there existsa constant 119862 independent of V such that
sup119873
10038171003817100381710038171003817100381710038171003817100381710038171003817
119873
sum
119895=0
V (119909119895) 119865
119895(119909)
100381710038171003817100381710038171003817100381710038171003817100381710038171198712120596120572120573(119868)
le 119862 max119909isin[minus11]
|V (119909)| (42)
where 119865119895(119909) 119895 = 0 1 119873 are the Lagrange interpolation
basis functions associated with the Jacobi collocation points119909
119895119873
119895=0
Lemma 8 (see [34]) For all measurable function 119891 ge 0 thefollowing generalized Hardyrsquos inequality
(int
119887
119886
1003816100381610038161003816(119879119891) (119909)1003816100381610038161003816
119902
119906 (119909) 119889119909)
1119902
le (int
119887
119886
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901
V (119909) 119889119909)1119901
(43)
6 Advances in Mathematical Physics
holds if and only if
sup119886lt119909lt119887
(int
119887
119909
119906 (119905) 119889119905)
1119902
(int
119909
119886
V1minus1199011015840
(119905) 119889119905)
11199011015840
lt infin 1199011015840
=119901
119901 minus 1
(44)
for the case 1 lt 119901 le 119902 lt infin Here 119879 is an operator of the form
(119879119865) (119909) = int
119909
119886
119896 (119909 119905) 119891 (119905) 119889119905 (45)
with 119896(119909 119905) a given kernel 119906 V are nonnegative weightfunctions and minusinfin le 119886 lt 119887 le infin
4 Convergence Analysis
This section is devoted to provide a convergence analysis forthe numerical scheme The goal is to show that the rate ofconvergence is exponential that is the spectral accuracy canbe obtained for the proposed approximations Firstly we willcarry our convergence analysis in 119871
infin space
Theorem 9 Let 119906(119909) be the exact solution of the fractionaldelay-integrodifferential equation (7)-(8) which is assumedto be sufficiently smooth Assume that 119880(119909) and 119880
120574
(119909) areobtained by using the spectral collocation scheme (26)-(27)together with a polynomial interpolation (25) If 120574 associatedwith the weakly singular kernel satisfies 0 lt 120574 lt 1 and119906 isin 119867
119898+1
120596minus120583minus120583(119868) then
1003817100381710038171003817119880120574
minus 119863120574
1199061003817100381710038171003817119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(46)
119880 minus 119906119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(47)
provided that 119873 is sufficiently large where 119862 is a constantindependent of 119873 but which will depend on the bounds of thefunctions 119870(119909 119904) and the index 120583
119870lowast
= max119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
+ max119909isin[minus11]
10038161003816100381610038161198702(119909 120578 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(48)
U =1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868) (49)
Proof We let
(119870 (119904 (119909119894 ])) 120601 (119904 (119909
119894 ])))
119873
=
119873
sum
119896=0
119870(119904 (119909119894 ]119896)) 120601 (119904 (119909
119894 ]119896)) 120596
00
119896
(50)
The numerical scheme (26)-(27) can be written as
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2(119870
1(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
119880 (120579 (119909minus120583minus120583
119894 ])))
119873
+
119902 (1 + 119909minus120583minus120583
119894)
2(119870
2(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 ]))
119880 (120578 (119909minus120583minus120583
119894 ])))
119873
(51)
119906119894=
1
Γ (120574)(119879
2)
120574
int
119909119894
minus1
(119909119894minus 120579)
minus120583
119880120574
(120579) 119889120579 + 119906minus1 (52)
which gives
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2int
1
minus1
1198701(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
times 119880 (120579 (119909minus120583minus120583
119894 ])) 119889] + 119868
1(119909
minus120583minus120583
119894)
+
119902 (1 + 119909minus120583minus120583
119894)
2int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585))
times 119880 (120578 (119909minus120583minus120583
119894 120585)) 119889120585 + 119868
2(119909
minus120583minus120583
119894)
(53)
where
1198681(119909) =
1 + 119909
2int
1
minus1
1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]
minus1 + 119909
2(119870
1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))
119873
1198682(119909) =
119902 (1 + 119909)
2int
1
minus1
1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585
minus119902 (1 + 119909)
2(119870
2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))
119873
(54)
Advances in Mathematical Physics 7
Using the integration error estimates from Jacobi-Gausspolynomials quadrature in Lemma 1 we have
10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
119880 (120579 (119909 sdot))1198712(119868)
le 119862119873minus119898
119870lowast
1119880
1198712(119868)
(55)
10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)
le 119862119873minus119898
119870lowast
2119880
1198712(119868)
(56)
119870lowast
1= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
119870lowast
2= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(57)
From (18) (53) can be rewritten as
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579)119880 (120579) 119889120579 + 119868
1(119909
minus120583minus120583
119894)
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578)119880 (120578) 119889120578
+ 1198682(119909
minus120583minus120583
119894)
(58)
Let 119890 and 119890120574 denote the error function
119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574
(119909) = 119880120574
(119909) minus 119906120574
(119909)
(59)
we have
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119906 (120579) 119889120579
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119890 (120579) 119889120579
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119906 (120578) 119889120578
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119890 (120578) 119889120578
+ 1198681(119909
minus120583minus120583
119894) + 119868
2(119909
minus120583minus120583
119894)
(60)
Multiplying119865119894(119909)onboth sides of (62) and (52) summing
up from 0 to119873 and using (7)-(8) yield
119880120574
(119909) = 119880 (119909) + 119868minus120583minus120583
119873(
119873
sum
119895=0
119906119895119865119895(119902119909 + 119902 minus 1))
+ 119868minus120583minus120583
119873119892 (119909)
+ 119868minus120583minus120583
119873[119863
120574
119906 (119909) minus 119906 (119909)
minus119906 (119902 (119909) + 119902 minus 1) minus 119892 (119909)]
+ 119868minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ 119868minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909)
(61)
119880 (119909) = 119868minus120583minus120583
119873(119906 (119909) minus 119906
minus1)
+ 119868minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
+ 119868minus120583minus120583
119873(119906
minus1)
(62)
where
1198691(119909) = 119868
minus120583minus120583
1198731198681(119909) 119869
2(119909) = 119868
minus120583minus120583
1198731198682(119909) (63)
From (61)-(62) we have
119890120574
(119909) = 119890 (119909) + int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(64)
119890 (119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+ 1198694(119909) + 119869
8(119909)
(65)
where
1198693(119909) = 119868
minus120583minus120583
119873119863120574
119906 (119909) minus 119863120574
119906 (119909)
1198694(119909) = 119868
minus120583minus120583
119873119906 (119909) minus 119906 (119909)
1198695(119909) = 119868
minus120583minus120583
119873119906 (119902119909 + 119902 minus 1) minus 119906 (119902119909 + 119902 minus 1)
1198696(119909) = 119868
minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
minus int
119909
minus1
119870 (119909 119904) 119890 (119904) 119889119904
8 Advances in Mathematical Physics
1198697(119909) = 119868
minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
minus int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1198698(119909) = 119868
minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
minus1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
(66)
Due to (64) we obtain
119890120574
(119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
times 119890120574
(119911) 119889119911 119889120578 + 119867 (119909)
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(67)
where
119867(119909) = int
119909
minus1
1198701(119909 120579) int
120579
minus1
(1198694(120579) + 119869
8(120579)) 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(1198694(120578) + 119869
8(120578)) 119889120578
+ 1198694(119909) + 119869
8(119909)
(68)
Using the Dirichletrsquos formula which sates that
int
119909
minus1
int
120579
minus1
Φ (120579 120591) 119889120591 119889120579 = int
119909
minus1
int
119909
120591
Φ (120579 120591) 119889120579 119889120591 (69)
provided that the integral exists we obtain
1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
=1
Γ (120574)int
119909
minus1
(int
119909
120579
1198701(119909 120579) (120579 minus 120591)
minus120583
119889120579) 119890120574
(120591) 119889120591
le1
Γ (120574)21minus120583 max
119909120591isin[minus11]
100381610038161003816100381610038161(119909 120591)
10038161003816100381610038161003816int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
≜ 1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
(70)
letting 120578 = 119902120591 + 119902 minus 1 119911 = 119902120579 + 119902 minus 1 we have
1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
119890120574
(119911) 119889119911 119889120578
=1
Γ (120574)119902int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
119902120591+119902minus1
minus1
(119909 minus 119911)minus120583
119890120574
(119911) 119889119911) 119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
120591
minus1
(1 minus 120579)minus120583
119890120574
(119902120579 + 119902 minus 1) 119889120579)119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
(int
119909
120579
1198702(119909 119902120591 + 119902 minus 1)
times (1 minus 120579)
minus120583
119889120591) 119890120574
times (119902120579 + 119902 minus 1) 119889120579
le1
Γ (120574)21minus120583
1199021+120574 max
119909120591isin[minus11]
10038161003816100381610038161198702(119909 120591)
1003816100381610038161003816
times int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
≜ 1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
(71)
Then (67) gives
119890120574
(119909) le1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
+1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
+ 119867 (119909) + 1198691(119909) + 119869
2(119909) + 119869
3(119909)
+ 1198694(119909) + 119869
5(119909) + 119869
6(119909) + 119869
7(119909)
(72)
It follows from the Gronwall inequality that
1003817100381710038171003817119890120574
(119909)1003817100381710038171003817119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(73)
From (65) we have
119890 (119909)119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(74)
Advances in Mathematical Physics 9
We now apply Lemma 7 to obtain that
100381710038171003817100381711986911003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
100381710038171003817100381711986921003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
(75)
Due to Lemma 2
100381710038171003817100381711986931003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus1198981003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log1198731003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986941003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(76)
By virtue of Lemma 2 (33b) with119898 = 1
100381710038171003817100381711986961003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
100381710038171003817100381711986971003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
(77)
We now estimate the term 1198698(119909) It follows from Lemmas
5 and 6 with 119870(119909 120591) = 1Γ(120574) that
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
12057410038171003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873M119890
120574
)10038171003817100381710038171003817119871infin(119868)
le (1 +10038171003817100381710038171003817119868minus120583minus120583
119873
10038171003817100381710038171003817119871infin(119868))119862119873
minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581
le
119862119873(12)minus120583minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus120581 log1198731003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
1
2le 120583 lt 1
(78)
where in the last step we have used Lemma 6 under thefollowing assumption
1
2minus 120583 lt 120581 lt 1 minus 120583 when 0 lt 120583 lt
1
2
0 lt 120581 lt 1 minus 120583 when 1
2le 120583 lt 1
(79)
Provided that119873 is sufficiently large combining (75) (76)(77) and (78) gives1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)1003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
119880 (119909) minus 119906 (119909)119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
(80)
Using 120574 = 1 minus 120583 we have the desired estimates (46) and (47)
Next we will give the error estimates in 1198712
120596minus120583minus120583 space
Theorem 10 If the hypotheses given in Theorem 9 hold then1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)10038171003817100381710038171198712120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
119880(119909) minus 119906(119909)1198712
120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
(81)
10 Advances in Mathematical Physics
for any 120581 isin (0 120574) provided that119873 is sufficiently large and 119862 isa constant independent of119873 where
1198811= 119870
lowast
(1199061198712(119868)
+1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868))
1198812= 119870
lowast
1199061198712(119868)
(82)
Proof Byusing the generalization ofGronwalls Lemma 4 andthe Hardy inequality Lemma 8 it follows from (72) that
100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
1198901198712
120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
(83)
Now using Lemma 7 we have10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
1(119906
1198712(119868)
+ 119890119871infin(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
2(119906
1198712(119868)
+ 119890119871infin(119868))
(84)
By the convergence result in Theorem 9 (119898 = 1) we have
119890119871infin(119868)
le 119862(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868)) (85)
So that10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
1(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
2(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
(86)
Due to Lemma 2 (33a)10038171003817100381710038171198693
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1198981003816100381610038161003816119863120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
(87)
By virtue of Lemma 2 (33a) with119898 = 1
1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
10038161003816100381610038161003816100381610038161003816
int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
100381610038161003816100381610038161003816100381610038161003816
int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
(88)
Finally it follows from Lemmas 5 and 7 that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
120574100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le10038171003817100381710038171003817119868minus120583minus120583
119873(M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
+1003817100381710038171003817M119890
120574
minusT11987311989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)
le 1198621003817100381710038171003817M119890
120574
minusT1198731198901205741003817100381710038171003817119871infin(119868)
le 119862119873minus1205811003817100381710038171003817M119890
12057410038171003817100381710038170120581
le 119862119873minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
(89)
where in the last step we used Lemma 6 for any 120581 isin (0 1minus120583)By the convergence result in Theorem 9 we obtain that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)
le
11986211987312minus120583minus119898minus120581
(119870lowast
1199061198712(119868)
+ 11987312U) 0 lt 120583 lt
1
2
119862119873minus119898minus120581 log119873(119870
lowast
1199061198712(119868)
+ 11987312U)
1
2le 120583 lt 1
(90)
for119873 sufficiently large and for any 120581 isin (0 1 minus 120583) The desiredestimates (81) follows from the previous estimates and (83)with 120574 = 1 minus 120583
5 Algorithm Implementation andNumerical Experiments
In this subsection we present the numerical results obtainedby the proposed collocation spectral method The estimatesin Section 4 indicates that the convergence of numericalsolutions is exponential if the exact solution is smooth Toconfirm the theoretical prediction a numerical experimentis carried out by considering the following example
Example 1 Consider the following fractional integrodifferen-tial equation
119863075
119910 (119905) =6119905225
Γ (325)minus1199052
119890119905
5119910 (119905) minus
(119902119905)2
5119910 (119902119905)
+ int
119905
0
119890119905
119904119910 (119904) 119889119904 + int
119902119905
0
120591119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 0
(91)
The corresponding exact solution is given by 119910(119905) = 1199053
Figure 1 presents the approximate and exact solutionson the left-hand side and presents the approximate and
Advances in Mathematical Physics 11
0 02 04 06 08 10
02
04
06
08
1
12
14
Exact solutionApproximate solution
(a)
0 02 04 06 08 10
05
1
15
2
25
Exact solutionApproximate solution
(b)
Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075
119910(119905) (b)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(a)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(b)
Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075
119910(119905) (b) versusthe number of collocation points in 119871
infin and 1198712
120596norms
exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871
infin and 1198712
120596minus120583minus120583 norms As
expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions
Example 2 Consider the following fractional integrodiffer-ential equation
119863120572
119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int
119905
0
119890119904(119905minus119904)
119910 (119904) 119889119904
+ int
119902119905
0
119902119905119890120591(119902119905minus120591)
119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 1
(92)
when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
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Stochastic AnalysisInternational Journal of
4 Advances in Mathematical Physics
Next using a (119873 + 1)-point Gauss quadrature formularelative to the Jacobi weight 12059600
119896119873
119894=0 the integration term in
(21) can be approximated by
int
1
minus1
1198701(119909
minus120583minus120583
120579 (119909minus120583minus120583
119894 ])) 119906 (120579 (119909minus120583minus120583
119894 ])) 119889]
=
119873
sum
119896=0
1198701(119909
119894 120579 (119909
119894 ]119896)) 119906 (120579 (119909
119894 ]119896)) 120596
00
119896
int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585)) 119906 (120578 (119909
minus120583minus120583
119894 120585)) 119889120585
=
119873
sum
119896=0
1198702(119909
119894 120578 (119909
119894 120585119896)) 119906 (120578 (119909
119894 120585119896)) 120596
00
119896
(23)
The sets ]119896119873
119896=0and 120585
119896119873
119896=0coincide with the Jacobi-Gauss
points corresponding Jacobi weights 12059600119896119873
119896=0 that is ]
119896119873
119896=0
and 120585119896119873
119896=0are Legendre-Gauss points
Using a (119873 + 1)-point Gauss quadrature formula relativeto the Jacobi weight 120596minus1205830 the integration term in (22) can beapproximated by
int
1
minus1
(1 minus ])120583119863120574
119906 (120579 (119909minus120583minus120583
119894 ])) 119889120579
=
119873
sum
119896=0
119863120574
119906 (120579 (119909minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896
(24)
where the set ]119896119873
119896=0is the Jacobi-Gauss points correspond-
ing to the weight 120596minus1205830We use 119906
119894 119906120574
119894to approximate the function value 119906(119909minus120583minus120583
119894)
119863120574
119906(119909minus120583minus120583
119894) 0 le 119894 le 119873 and expand 119906 and 119906120574 using Lagrange
interpolation polynomials that is
119880 (119909) =
119873
sum
119895=0
119906119895119865119895(119909) 119880
120574
(119909) =
119873
sum
119895=0
119906120574
119895119865119895(119909) (25)
where 119865119895(119909) is the Lagrange interpolation basis function
associated with 119909minus120583minus120583
119894119873
119894=0which is the set of (119873 + 1)
Jacobi-Gauss points The Jacobi collocation methods are toseek 119906
120574
119894119873
119894=0and 119906
119894119873
119894=0such that the following collocation
equations hold
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1))
+ 119892 (119909minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2
119873
sum
119895=0
119906119895(
119873
sum
119896=0
1198701(119909
minus120583minus120583
119894 120579 (119909
119894 ]119896))
times119865119895(120579 (119909
minus120583minus120583
119894 ]119896)) 120596
00
119896)
+
119902 (1 + 119909minus120583minus120583
119894)
2
119873
sum
119895=0
119906119895(
119873
sum
119896=0
1198702(119909
minus120583minus120583
119894 120578 (119909
119894 ]119896))
times 119865119895(120578 (119909
minus120583minus120583
119894 ]119896)) 120596
00
119896)
(26)
119906119894=
1
Γ (120574)(
(1 + 119909minus120583minus120583
119894) 119879
4)
120574
119873
sum
119895=0
119906120574
119895
times (
119873
sum
119896=0
119865119895(119904 (119909
minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896) + 119906 (minus1)
(27)
Writing 119880119873
= (1199060 119906
1 119906
119873)119879 and 119880
120574
119873= (119906
120574
0 119906
120574
1
119906120574
119873)119879 we obtain the following of the matrix form from (26)-
(27)
119880120574
119873= (119864 + 119871 + 119862 + 119863)119880
119873+ 119866
119880119873= 119880
minus1+ 119861119880
120574
119873
(28)
where
119864 is the identity matrix
119866 = (119892 (119909minus120583minus120583
0) 119892 (119909
minus120583minus120583
1) 119892 (119909
minus120583minus120583
119873))
119879
119871119894119895= 119865
119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)
119862119894119895=1 + 119909
minus120583minus120583
119894
2
119873
sum
119896=0
1(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]119896)) 119865
119895
times (120579 (119909119894 ]119896)) 120596
00
119896
119863119894119895=
119902 (1 + 119909minus120583minus120583
119894)
2
119873
sum
119896=0
2(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 ]119896)) 119865
119895
times (120578 (119909minus120583minus120583
119894 ]119896)) 120596
00
119896
119861119894119895=
1
Γ (120574)(
(1 + 119909minus120583minus120583
119894) 119879
4)
120574
times
119873
sum
119896=0
119865119895(119904 (119909
minus120583minus120583
119894 ]119896)) 120596
minus1205830
119896
119880minus1
= (119906 (minus1) 119906 (minus1) 119906 (minus1))119879
(29)
We can get the values of 119906119894119873
119894=0and 119906
120574
119894119873
119894=0by solving the
system of linear system (28) Therefore the expressions of119880(119909) and 119880
120574
(119909) can be obtained
3 Some Useful Lemmas
In this section we will provide some elementary lemmaswhich are important for the derivation of the main results inthe subsequent section Let 119868 = (minus1 1)
Advances in Mathematical Physics 5
Lemma 1 (see [27]) Assume that an (119873 + 1)-point Gaussquadrature formula relative to the Jacobi weight is used tointegrate the product 119906120593 where 119906 isin 119867
119898
(119868)with 119868 for some119898 ge
1 and 120593 isin P119873 Then there exists a constant 119862 independent of
N such that100381610038161003816100381610038161003816100381610038161003816
int
1
minus1
119906 (119909) 120593 (119909) 119889119909 minus (119906 120593)119873
100381610038161003816100381610038161003816100381610038161003816
le 119862119873minus119898
|119906|119867119898119873
120596120572120573(119868)
100381710038171003817100381712059310038171003817100381710038171198712120596120572120573(119868)
(30)
where
|119906|119867119898119873
120596120572120573(119868)
= (
119898
sum
119895=min(119898119873+1)
10038171003817100381710038171003817119906(119895)10038171003817100381710038171003817
2
1198712
120596120572120573(119868)
)
12
(119906 120593)119873=
119873
sum
119895=0
119906 (119909119895) 120593 (119909
119895) 120596
119895
(31)
Lemma 2 (see [26 27]) Assume that 119906 isin 119867119898119873
120596minus120583minus120583
(119868) anddenote by 119868minus120583minus120583
119873119906 its interpolation polynomial associated with
the (119873 + 1) Jacobi-Gauss points 119909119895119873
119895=0 namely
119868minus120583minus120583
119873119906 =
119873
sum
119894=0
119906 (119909119894) 119865 (119909
119894) (32)
Then the following estimates hold
10038171003817100381710038171003817119906 minus 119868
minus120583minus120583
119873119906100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
|119906|119867119898119873
120596120572120573(119868) (33a)
10038171003817100381710038171003817119906 minus 119868
minus120583minus120583
11987311990610038171003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 le 120583 lt
1
2
11986211987312minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(33b)
where 120596119888 = 120596minus12minus12 denotes the Chebyshev weight function
Lemma 3 (see [28]) Assume that 119865119895(119909)
119873
119895=0are the 119873-th
degree Lagrange basis polynomials associated with the Gausspoints of the Jacobi polynomials Then
100381710038171003817100381710038171003817119868120572120573
119873
100381710038171003817100381710038171003817119871infin(119868)le max
119909isin[minus11]
119873
sum
119895=0
10038161003816100381610038161003816119865119895(119909)
10038161003816100381610038161003816
=
O (log119873) minus1 lt 120572 120573 le minus1
2
O (119873120574+12
) 120574 = max (120572 120573) 119900119905ℎ119890119903119908119894119904119890(34)
Lemma 4 (Gronwall inequality see [29] Lemma 711)Suppose that 119871 ge 0 0 lt 120583 lt 1 and 119906 and V are a nonnegativelocally integrable functions defined on [minus1 1] satisfying
119906 (119909) le V (119909) + 119871int
119909
minus1
(119909 minus 120591)minus120583
119906 (120591) 119889120591 (35)
Then there exists a constant 119862 = 119862(120583) such that
119906 (119909) le V (119909) + 119862119871int
119909
minus1
(119909 minus 120591)minus120583V (120591) 119889120591 119891119900119903 minus 1 le 119909 lt 1
(36)
Lemma 5 (see [30 31]) For a nonnegative integer 119903 and 120581 isin
(0 1) there exists a constant119862119903120581
gt 0 such that for any functionV isin 119862
119903120581
([minus1 1]) there exists a polynomial function T119873V isin
P119873such that
1003817100381710038171003817V minusT119873V1003817100381710038171003817119871infin(119868) le 119862
119903120581119873minus(119903+120581)
V119903120581 (37)
where sdot 119903120581
is the standard norm in 119862119903120581
([minus1 1]) which isdenoted by the space of functions whose 119903th derivatives areHolder continuous with exponent 120581 endowed with the usualnorm
V119903120581
= max0le120581le119903
max119909isin[minus11]
1003816100381610038161003816120597120581
119909V (119909)1003816100381610038161003816
+ max0le120581le119903
sup119909119910isin[minus11]119909 = 119910
1003816100381610038161003816120597120581
119909V (119909) minus 120597
120581
119909V (119909)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
120581
(38)
T119873is a linear operator from 119862
119903120581
([minus1 1]) intoP119873
Lemma 6 (see [32]) Let 120581 isin (0 1) and letM be defined by
(MV) (119909) = int
119909
minus1
(119909 minus 120591)minus120583
119870 (119909 120591) V (120591) 119889120591 (39)
Then for any function V isin 119862([minus1 1]) there exists a positiveconstant 119862 such that
10038161003816100381610038161003816MV (1199091015840) minusMV (11990910158401015840)
1003816100381610038161003816100381610038161003816100381610038161199091015840 minus 11990910158401015840
1003816100381610038161003816
le 119862 max119909isin[minus11]
|V (119909)| (40)
under the assumption that 0 lt 120581 lt 1 minus 120583 for any 1199091015840
11990910158401015840
isin
[minus1 1] and 1199091015840 = 11990910158401015840 This implies that
MV0120581
le 119862 max119909isin[minus11]
|V (119909)| 0 lt 120581 lt 1 minus 120583 (41)
Lemma7 (see [33]) For every bounded function V there existsa constant 119862 independent of V such that
sup119873
10038171003817100381710038171003817100381710038171003817100381710038171003817
119873
sum
119895=0
V (119909119895) 119865
119895(119909)
100381710038171003817100381710038171003817100381710038171003817100381710038171198712120596120572120573(119868)
le 119862 max119909isin[minus11]
|V (119909)| (42)
where 119865119895(119909) 119895 = 0 1 119873 are the Lagrange interpolation
basis functions associated with the Jacobi collocation points119909
119895119873
119895=0
Lemma 8 (see [34]) For all measurable function 119891 ge 0 thefollowing generalized Hardyrsquos inequality
(int
119887
119886
1003816100381610038161003816(119879119891) (119909)1003816100381610038161003816
119902
119906 (119909) 119889119909)
1119902
le (int
119887
119886
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901
V (119909) 119889119909)1119901
(43)
6 Advances in Mathematical Physics
holds if and only if
sup119886lt119909lt119887
(int
119887
119909
119906 (119905) 119889119905)
1119902
(int
119909
119886
V1minus1199011015840
(119905) 119889119905)
11199011015840
lt infin 1199011015840
=119901
119901 minus 1
(44)
for the case 1 lt 119901 le 119902 lt infin Here 119879 is an operator of the form
(119879119865) (119909) = int
119909
119886
119896 (119909 119905) 119891 (119905) 119889119905 (45)
with 119896(119909 119905) a given kernel 119906 V are nonnegative weightfunctions and minusinfin le 119886 lt 119887 le infin
4 Convergence Analysis
This section is devoted to provide a convergence analysis forthe numerical scheme The goal is to show that the rate ofconvergence is exponential that is the spectral accuracy canbe obtained for the proposed approximations Firstly we willcarry our convergence analysis in 119871
infin space
Theorem 9 Let 119906(119909) be the exact solution of the fractionaldelay-integrodifferential equation (7)-(8) which is assumedto be sufficiently smooth Assume that 119880(119909) and 119880
120574
(119909) areobtained by using the spectral collocation scheme (26)-(27)together with a polynomial interpolation (25) If 120574 associatedwith the weakly singular kernel satisfies 0 lt 120574 lt 1 and119906 isin 119867
119898+1
120596minus120583minus120583(119868) then
1003817100381710038171003817119880120574
minus 119863120574
1199061003817100381710038171003817119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(46)
119880 minus 119906119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(47)
provided that 119873 is sufficiently large where 119862 is a constantindependent of 119873 but which will depend on the bounds of thefunctions 119870(119909 119904) and the index 120583
119870lowast
= max119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
+ max119909isin[minus11]
10038161003816100381610038161198702(119909 120578 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(48)
U =1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868) (49)
Proof We let
(119870 (119904 (119909119894 ])) 120601 (119904 (119909
119894 ])))
119873
=
119873
sum
119896=0
119870(119904 (119909119894 ]119896)) 120601 (119904 (119909
119894 ]119896)) 120596
00
119896
(50)
The numerical scheme (26)-(27) can be written as
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2(119870
1(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
119880 (120579 (119909minus120583minus120583
119894 ])))
119873
+
119902 (1 + 119909minus120583minus120583
119894)
2(119870
2(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 ]))
119880 (120578 (119909minus120583minus120583
119894 ])))
119873
(51)
119906119894=
1
Γ (120574)(119879
2)
120574
int
119909119894
minus1
(119909119894minus 120579)
minus120583
119880120574
(120579) 119889120579 + 119906minus1 (52)
which gives
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2int
1
minus1
1198701(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
times 119880 (120579 (119909minus120583minus120583
119894 ])) 119889] + 119868
1(119909
minus120583minus120583
119894)
+
119902 (1 + 119909minus120583minus120583
119894)
2int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585))
times 119880 (120578 (119909minus120583minus120583
119894 120585)) 119889120585 + 119868
2(119909
minus120583minus120583
119894)
(53)
where
1198681(119909) =
1 + 119909
2int
1
minus1
1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]
minus1 + 119909
2(119870
1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))
119873
1198682(119909) =
119902 (1 + 119909)
2int
1
minus1
1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585
minus119902 (1 + 119909)
2(119870
2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))
119873
(54)
Advances in Mathematical Physics 7
Using the integration error estimates from Jacobi-Gausspolynomials quadrature in Lemma 1 we have
10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
119880 (120579 (119909 sdot))1198712(119868)
le 119862119873minus119898
119870lowast
1119880
1198712(119868)
(55)
10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)
le 119862119873minus119898
119870lowast
2119880
1198712(119868)
(56)
119870lowast
1= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
119870lowast
2= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(57)
From (18) (53) can be rewritten as
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579)119880 (120579) 119889120579 + 119868
1(119909
minus120583minus120583
119894)
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578)119880 (120578) 119889120578
+ 1198682(119909
minus120583minus120583
119894)
(58)
Let 119890 and 119890120574 denote the error function
119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574
(119909) = 119880120574
(119909) minus 119906120574
(119909)
(59)
we have
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119906 (120579) 119889120579
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119890 (120579) 119889120579
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119906 (120578) 119889120578
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119890 (120578) 119889120578
+ 1198681(119909
minus120583minus120583
119894) + 119868
2(119909
minus120583minus120583
119894)
(60)
Multiplying119865119894(119909)onboth sides of (62) and (52) summing
up from 0 to119873 and using (7)-(8) yield
119880120574
(119909) = 119880 (119909) + 119868minus120583minus120583
119873(
119873
sum
119895=0
119906119895119865119895(119902119909 + 119902 minus 1))
+ 119868minus120583minus120583
119873119892 (119909)
+ 119868minus120583minus120583
119873[119863
120574
119906 (119909) minus 119906 (119909)
minus119906 (119902 (119909) + 119902 minus 1) minus 119892 (119909)]
+ 119868minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ 119868minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909)
(61)
119880 (119909) = 119868minus120583minus120583
119873(119906 (119909) minus 119906
minus1)
+ 119868minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
+ 119868minus120583minus120583
119873(119906
minus1)
(62)
where
1198691(119909) = 119868
minus120583minus120583
1198731198681(119909) 119869
2(119909) = 119868
minus120583minus120583
1198731198682(119909) (63)
From (61)-(62) we have
119890120574
(119909) = 119890 (119909) + int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(64)
119890 (119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+ 1198694(119909) + 119869
8(119909)
(65)
where
1198693(119909) = 119868
minus120583minus120583
119873119863120574
119906 (119909) minus 119863120574
119906 (119909)
1198694(119909) = 119868
minus120583minus120583
119873119906 (119909) minus 119906 (119909)
1198695(119909) = 119868
minus120583minus120583
119873119906 (119902119909 + 119902 minus 1) minus 119906 (119902119909 + 119902 minus 1)
1198696(119909) = 119868
minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
minus int
119909
minus1
119870 (119909 119904) 119890 (119904) 119889119904
8 Advances in Mathematical Physics
1198697(119909) = 119868
minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
minus int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1198698(119909) = 119868
minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
minus1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
(66)
Due to (64) we obtain
119890120574
(119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
times 119890120574
(119911) 119889119911 119889120578 + 119867 (119909)
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(67)
where
119867(119909) = int
119909
minus1
1198701(119909 120579) int
120579
minus1
(1198694(120579) + 119869
8(120579)) 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(1198694(120578) + 119869
8(120578)) 119889120578
+ 1198694(119909) + 119869
8(119909)
(68)
Using the Dirichletrsquos formula which sates that
int
119909
minus1
int
120579
minus1
Φ (120579 120591) 119889120591 119889120579 = int
119909
minus1
int
119909
120591
Φ (120579 120591) 119889120579 119889120591 (69)
provided that the integral exists we obtain
1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
=1
Γ (120574)int
119909
minus1
(int
119909
120579
1198701(119909 120579) (120579 minus 120591)
minus120583
119889120579) 119890120574
(120591) 119889120591
le1
Γ (120574)21minus120583 max
119909120591isin[minus11]
100381610038161003816100381610038161(119909 120591)
10038161003816100381610038161003816int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
≜ 1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
(70)
letting 120578 = 119902120591 + 119902 minus 1 119911 = 119902120579 + 119902 minus 1 we have
1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
119890120574
(119911) 119889119911 119889120578
=1
Γ (120574)119902int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
119902120591+119902minus1
minus1
(119909 minus 119911)minus120583
119890120574
(119911) 119889119911) 119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
120591
minus1
(1 minus 120579)minus120583
119890120574
(119902120579 + 119902 minus 1) 119889120579)119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
(int
119909
120579
1198702(119909 119902120591 + 119902 minus 1)
times (1 minus 120579)
minus120583
119889120591) 119890120574
times (119902120579 + 119902 minus 1) 119889120579
le1
Γ (120574)21minus120583
1199021+120574 max
119909120591isin[minus11]
10038161003816100381610038161198702(119909 120591)
1003816100381610038161003816
times int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
≜ 1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
(71)
Then (67) gives
119890120574
(119909) le1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
+1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
+ 119867 (119909) + 1198691(119909) + 119869
2(119909) + 119869
3(119909)
+ 1198694(119909) + 119869
5(119909) + 119869
6(119909) + 119869
7(119909)
(72)
It follows from the Gronwall inequality that
1003817100381710038171003817119890120574
(119909)1003817100381710038171003817119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(73)
From (65) we have
119890 (119909)119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(74)
Advances in Mathematical Physics 9
We now apply Lemma 7 to obtain that
100381710038171003817100381711986911003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
100381710038171003817100381711986921003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
(75)
Due to Lemma 2
100381710038171003817100381711986931003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus1198981003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log1198731003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986941003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(76)
By virtue of Lemma 2 (33b) with119898 = 1
100381710038171003817100381711986961003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
100381710038171003817100381711986971003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
(77)
We now estimate the term 1198698(119909) It follows from Lemmas
5 and 6 with 119870(119909 120591) = 1Γ(120574) that
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
12057410038171003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873M119890
120574
)10038171003817100381710038171003817119871infin(119868)
le (1 +10038171003817100381710038171003817119868minus120583minus120583
119873
10038171003817100381710038171003817119871infin(119868))119862119873
minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581
le
119862119873(12)minus120583minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus120581 log1198731003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
1
2le 120583 lt 1
(78)
where in the last step we have used Lemma 6 under thefollowing assumption
1
2minus 120583 lt 120581 lt 1 minus 120583 when 0 lt 120583 lt
1
2
0 lt 120581 lt 1 minus 120583 when 1
2le 120583 lt 1
(79)
Provided that119873 is sufficiently large combining (75) (76)(77) and (78) gives1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)1003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
119880 (119909) minus 119906 (119909)119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
(80)
Using 120574 = 1 minus 120583 we have the desired estimates (46) and (47)
Next we will give the error estimates in 1198712
120596minus120583minus120583 space
Theorem 10 If the hypotheses given in Theorem 9 hold then1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)10038171003817100381710038171198712120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
119880(119909) minus 119906(119909)1198712
120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
(81)
10 Advances in Mathematical Physics
for any 120581 isin (0 120574) provided that119873 is sufficiently large and 119862 isa constant independent of119873 where
1198811= 119870
lowast
(1199061198712(119868)
+1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868))
1198812= 119870
lowast
1199061198712(119868)
(82)
Proof Byusing the generalization ofGronwalls Lemma 4 andthe Hardy inequality Lemma 8 it follows from (72) that
100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
1198901198712
120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
(83)
Now using Lemma 7 we have10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
1(119906
1198712(119868)
+ 119890119871infin(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
2(119906
1198712(119868)
+ 119890119871infin(119868))
(84)
By the convergence result in Theorem 9 (119898 = 1) we have
119890119871infin(119868)
le 119862(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868)) (85)
So that10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
1(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
2(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
(86)
Due to Lemma 2 (33a)10038171003817100381710038171198693
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1198981003816100381610038161003816119863120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
(87)
By virtue of Lemma 2 (33a) with119898 = 1
1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
10038161003816100381610038161003816100381610038161003816
int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
100381610038161003816100381610038161003816100381610038161003816
int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
(88)
Finally it follows from Lemmas 5 and 7 that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
120574100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le10038171003817100381710038171003817119868minus120583minus120583
119873(M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
+1003817100381710038171003817M119890
120574
minusT11987311989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)
le 1198621003817100381710038171003817M119890
120574
minusT1198731198901205741003817100381710038171003817119871infin(119868)
le 119862119873minus1205811003817100381710038171003817M119890
12057410038171003817100381710038170120581
le 119862119873minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
(89)
where in the last step we used Lemma 6 for any 120581 isin (0 1minus120583)By the convergence result in Theorem 9 we obtain that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)
le
11986211987312minus120583minus119898minus120581
(119870lowast
1199061198712(119868)
+ 11987312U) 0 lt 120583 lt
1
2
119862119873minus119898minus120581 log119873(119870
lowast
1199061198712(119868)
+ 11987312U)
1
2le 120583 lt 1
(90)
for119873 sufficiently large and for any 120581 isin (0 1 minus 120583) The desiredestimates (81) follows from the previous estimates and (83)with 120574 = 1 minus 120583
5 Algorithm Implementation andNumerical Experiments
In this subsection we present the numerical results obtainedby the proposed collocation spectral method The estimatesin Section 4 indicates that the convergence of numericalsolutions is exponential if the exact solution is smooth Toconfirm the theoretical prediction a numerical experimentis carried out by considering the following example
Example 1 Consider the following fractional integrodifferen-tial equation
119863075
119910 (119905) =6119905225
Γ (325)minus1199052
119890119905
5119910 (119905) minus
(119902119905)2
5119910 (119902119905)
+ int
119905
0
119890119905
119904119910 (119904) 119889119904 + int
119902119905
0
120591119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 0
(91)
The corresponding exact solution is given by 119910(119905) = 1199053
Figure 1 presents the approximate and exact solutionson the left-hand side and presents the approximate and
Advances in Mathematical Physics 11
0 02 04 06 08 10
02
04
06
08
1
12
14
Exact solutionApproximate solution
(a)
0 02 04 06 08 10
05
1
15
2
25
Exact solutionApproximate solution
(b)
Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075
119910(119905) (b)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(a)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(b)
Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075
119910(119905) (b) versusthe number of collocation points in 119871
infin and 1198712
120596norms
exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871
infin and 1198712
120596minus120583minus120583 norms As
expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions
Example 2 Consider the following fractional integrodiffer-ential equation
119863120572
119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int
119905
0
119890119904(119905minus119904)
119910 (119904) 119889119904
+ int
119902119905
0
119902119905119890120591(119902119905minus120591)
119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 1
(92)
when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 5
Lemma 1 (see [27]) Assume that an (119873 + 1)-point Gaussquadrature formula relative to the Jacobi weight is used tointegrate the product 119906120593 where 119906 isin 119867
119898
(119868)with 119868 for some119898 ge
1 and 120593 isin P119873 Then there exists a constant 119862 independent of
N such that100381610038161003816100381610038161003816100381610038161003816
int
1
minus1
119906 (119909) 120593 (119909) 119889119909 minus (119906 120593)119873
100381610038161003816100381610038161003816100381610038161003816
le 119862119873minus119898
|119906|119867119898119873
120596120572120573(119868)
100381710038171003817100381712059310038171003817100381710038171198712120596120572120573(119868)
(30)
where
|119906|119867119898119873
120596120572120573(119868)
= (
119898
sum
119895=min(119898119873+1)
10038171003817100381710038171003817119906(119895)10038171003817100381710038171003817
2
1198712
120596120572120573(119868)
)
12
(119906 120593)119873=
119873
sum
119895=0
119906 (119909119895) 120593 (119909
119895) 120596
119895
(31)
Lemma 2 (see [26 27]) Assume that 119906 isin 119867119898119873
120596minus120583minus120583
(119868) anddenote by 119868minus120583minus120583
119873119906 its interpolation polynomial associated with
the (119873 + 1) Jacobi-Gauss points 119909119895119873
119895=0 namely
119868minus120583minus120583
119873119906 =
119873
sum
119894=0
119906 (119909119894) 119865 (119909
119894) (32)
Then the following estimates hold
10038171003817100381710038171003817119906 minus 119868
minus120583minus120583
119873119906100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
|119906|119867119898119873
120596120572120573(119868) (33a)
10038171003817100381710038171003817119906 minus 119868
minus120583minus120583
11987311990610038171003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 le 120583 lt
1
2
11986211987312minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(33b)
where 120596119888 = 120596minus12minus12 denotes the Chebyshev weight function
Lemma 3 (see [28]) Assume that 119865119895(119909)
119873
119895=0are the 119873-th
degree Lagrange basis polynomials associated with the Gausspoints of the Jacobi polynomials Then
100381710038171003817100381710038171003817119868120572120573
119873
100381710038171003817100381710038171003817119871infin(119868)le max
119909isin[minus11]
119873
sum
119895=0
10038161003816100381610038161003816119865119895(119909)
10038161003816100381610038161003816
=
O (log119873) minus1 lt 120572 120573 le minus1
2
O (119873120574+12
) 120574 = max (120572 120573) 119900119905ℎ119890119903119908119894119904119890(34)
Lemma 4 (Gronwall inequality see [29] Lemma 711)Suppose that 119871 ge 0 0 lt 120583 lt 1 and 119906 and V are a nonnegativelocally integrable functions defined on [minus1 1] satisfying
119906 (119909) le V (119909) + 119871int
119909
minus1
(119909 minus 120591)minus120583
119906 (120591) 119889120591 (35)
Then there exists a constant 119862 = 119862(120583) such that
119906 (119909) le V (119909) + 119862119871int
119909
minus1
(119909 minus 120591)minus120583V (120591) 119889120591 119891119900119903 minus 1 le 119909 lt 1
(36)
Lemma 5 (see [30 31]) For a nonnegative integer 119903 and 120581 isin
(0 1) there exists a constant119862119903120581
gt 0 such that for any functionV isin 119862
119903120581
([minus1 1]) there exists a polynomial function T119873V isin
P119873such that
1003817100381710038171003817V minusT119873V1003817100381710038171003817119871infin(119868) le 119862
119903120581119873minus(119903+120581)
V119903120581 (37)
where sdot 119903120581
is the standard norm in 119862119903120581
([minus1 1]) which isdenoted by the space of functions whose 119903th derivatives areHolder continuous with exponent 120581 endowed with the usualnorm
V119903120581
= max0le120581le119903
max119909isin[minus11]
1003816100381610038161003816120597120581
119909V (119909)1003816100381610038161003816
+ max0le120581le119903
sup119909119910isin[minus11]119909 = 119910
1003816100381610038161003816120597120581
119909V (119909) minus 120597
120581
119909V (119909)1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
120581
(38)
T119873is a linear operator from 119862
119903120581
([minus1 1]) intoP119873
Lemma 6 (see [32]) Let 120581 isin (0 1) and letM be defined by
(MV) (119909) = int
119909
minus1
(119909 minus 120591)minus120583
119870 (119909 120591) V (120591) 119889120591 (39)
Then for any function V isin 119862([minus1 1]) there exists a positiveconstant 119862 such that
10038161003816100381610038161003816MV (1199091015840) minusMV (11990910158401015840)
1003816100381610038161003816100381610038161003816100381610038161199091015840 minus 11990910158401015840
1003816100381610038161003816
le 119862 max119909isin[minus11]
|V (119909)| (40)
under the assumption that 0 lt 120581 lt 1 minus 120583 for any 1199091015840
11990910158401015840
isin
[minus1 1] and 1199091015840 = 11990910158401015840 This implies that
MV0120581
le 119862 max119909isin[minus11]
|V (119909)| 0 lt 120581 lt 1 minus 120583 (41)
Lemma7 (see [33]) For every bounded function V there existsa constant 119862 independent of V such that
sup119873
10038171003817100381710038171003817100381710038171003817100381710038171003817
119873
sum
119895=0
V (119909119895) 119865
119895(119909)
100381710038171003817100381710038171003817100381710038171003817100381710038171198712120596120572120573(119868)
le 119862 max119909isin[minus11]
|V (119909)| (42)
where 119865119895(119909) 119895 = 0 1 119873 are the Lagrange interpolation
basis functions associated with the Jacobi collocation points119909
119895119873
119895=0
Lemma 8 (see [34]) For all measurable function 119891 ge 0 thefollowing generalized Hardyrsquos inequality
(int
119887
119886
1003816100381610038161003816(119879119891) (119909)1003816100381610038161003816
119902
119906 (119909) 119889119909)
1119902
le (int
119887
119886
1003816100381610038161003816119891 (119909)1003816100381610038161003816
119901
V (119909) 119889119909)1119901
(43)
6 Advances in Mathematical Physics
holds if and only if
sup119886lt119909lt119887
(int
119887
119909
119906 (119905) 119889119905)
1119902
(int
119909
119886
V1minus1199011015840
(119905) 119889119905)
11199011015840
lt infin 1199011015840
=119901
119901 minus 1
(44)
for the case 1 lt 119901 le 119902 lt infin Here 119879 is an operator of the form
(119879119865) (119909) = int
119909
119886
119896 (119909 119905) 119891 (119905) 119889119905 (45)
with 119896(119909 119905) a given kernel 119906 V are nonnegative weightfunctions and minusinfin le 119886 lt 119887 le infin
4 Convergence Analysis
This section is devoted to provide a convergence analysis forthe numerical scheme The goal is to show that the rate ofconvergence is exponential that is the spectral accuracy canbe obtained for the proposed approximations Firstly we willcarry our convergence analysis in 119871
infin space
Theorem 9 Let 119906(119909) be the exact solution of the fractionaldelay-integrodifferential equation (7)-(8) which is assumedto be sufficiently smooth Assume that 119880(119909) and 119880
120574
(119909) areobtained by using the spectral collocation scheme (26)-(27)together with a polynomial interpolation (25) If 120574 associatedwith the weakly singular kernel satisfies 0 lt 120574 lt 1 and119906 isin 119867
119898+1
120596minus120583minus120583(119868) then
1003817100381710038171003817119880120574
minus 119863120574
1199061003817100381710038171003817119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(46)
119880 minus 119906119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(47)
provided that 119873 is sufficiently large where 119862 is a constantindependent of 119873 but which will depend on the bounds of thefunctions 119870(119909 119904) and the index 120583
119870lowast
= max119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
+ max119909isin[minus11]
10038161003816100381610038161198702(119909 120578 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(48)
U =1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868) (49)
Proof We let
(119870 (119904 (119909119894 ])) 120601 (119904 (119909
119894 ])))
119873
=
119873
sum
119896=0
119870(119904 (119909119894 ]119896)) 120601 (119904 (119909
119894 ]119896)) 120596
00
119896
(50)
The numerical scheme (26)-(27) can be written as
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2(119870
1(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
119880 (120579 (119909minus120583minus120583
119894 ])))
119873
+
119902 (1 + 119909minus120583minus120583
119894)
2(119870
2(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 ]))
119880 (120578 (119909minus120583minus120583
119894 ])))
119873
(51)
119906119894=
1
Γ (120574)(119879
2)
120574
int
119909119894
minus1
(119909119894minus 120579)
minus120583
119880120574
(120579) 119889120579 + 119906minus1 (52)
which gives
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2int
1
minus1
1198701(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
times 119880 (120579 (119909minus120583minus120583
119894 ])) 119889] + 119868
1(119909
minus120583minus120583
119894)
+
119902 (1 + 119909minus120583minus120583
119894)
2int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585))
times 119880 (120578 (119909minus120583minus120583
119894 120585)) 119889120585 + 119868
2(119909
minus120583minus120583
119894)
(53)
where
1198681(119909) =
1 + 119909
2int
1
minus1
1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]
minus1 + 119909
2(119870
1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))
119873
1198682(119909) =
119902 (1 + 119909)
2int
1
minus1
1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585
minus119902 (1 + 119909)
2(119870
2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))
119873
(54)
Advances in Mathematical Physics 7
Using the integration error estimates from Jacobi-Gausspolynomials quadrature in Lemma 1 we have
10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
119880 (120579 (119909 sdot))1198712(119868)
le 119862119873minus119898
119870lowast
1119880
1198712(119868)
(55)
10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)
le 119862119873minus119898
119870lowast
2119880
1198712(119868)
(56)
119870lowast
1= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
119870lowast
2= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(57)
From (18) (53) can be rewritten as
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579)119880 (120579) 119889120579 + 119868
1(119909
minus120583minus120583
119894)
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578)119880 (120578) 119889120578
+ 1198682(119909
minus120583minus120583
119894)
(58)
Let 119890 and 119890120574 denote the error function
119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574
(119909) = 119880120574
(119909) minus 119906120574
(119909)
(59)
we have
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119906 (120579) 119889120579
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119890 (120579) 119889120579
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119906 (120578) 119889120578
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119890 (120578) 119889120578
+ 1198681(119909
minus120583minus120583
119894) + 119868
2(119909
minus120583minus120583
119894)
(60)
Multiplying119865119894(119909)onboth sides of (62) and (52) summing
up from 0 to119873 and using (7)-(8) yield
119880120574
(119909) = 119880 (119909) + 119868minus120583minus120583
119873(
119873
sum
119895=0
119906119895119865119895(119902119909 + 119902 minus 1))
+ 119868minus120583minus120583
119873119892 (119909)
+ 119868minus120583minus120583
119873[119863
120574
119906 (119909) minus 119906 (119909)
minus119906 (119902 (119909) + 119902 minus 1) minus 119892 (119909)]
+ 119868minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ 119868minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909)
(61)
119880 (119909) = 119868minus120583minus120583
119873(119906 (119909) minus 119906
minus1)
+ 119868minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
+ 119868minus120583minus120583
119873(119906
minus1)
(62)
where
1198691(119909) = 119868
minus120583minus120583
1198731198681(119909) 119869
2(119909) = 119868
minus120583minus120583
1198731198682(119909) (63)
From (61)-(62) we have
119890120574
(119909) = 119890 (119909) + int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(64)
119890 (119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+ 1198694(119909) + 119869
8(119909)
(65)
where
1198693(119909) = 119868
minus120583minus120583
119873119863120574
119906 (119909) minus 119863120574
119906 (119909)
1198694(119909) = 119868
minus120583minus120583
119873119906 (119909) minus 119906 (119909)
1198695(119909) = 119868
minus120583minus120583
119873119906 (119902119909 + 119902 minus 1) minus 119906 (119902119909 + 119902 minus 1)
1198696(119909) = 119868
minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
minus int
119909
minus1
119870 (119909 119904) 119890 (119904) 119889119904
8 Advances in Mathematical Physics
1198697(119909) = 119868
minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
minus int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1198698(119909) = 119868
minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
minus1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
(66)
Due to (64) we obtain
119890120574
(119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
times 119890120574
(119911) 119889119911 119889120578 + 119867 (119909)
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(67)
where
119867(119909) = int
119909
minus1
1198701(119909 120579) int
120579
minus1
(1198694(120579) + 119869
8(120579)) 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(1198694(120578) + 119869
8(120578)) 119889120578
+ 1198694(119909) + 119869
8(119909)
(68)
Using the Dirichletrsquos formula which sates that
int
119909
minus1
int
120579
minus1
Φ (120579 120591) 119889120591 119889120579 = int
119909
minus1
int
119909
120591
Φ (120579 120591) 119889120579 119889120591 (69)
provided that the integral exists we obtain
1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
=1
Γ (120574)int
119909
minus1
(int
119909
120579
1198701(119909 120579) (120579 minus 120591)
minus120583
119889120579) 119890120574
(120591) 119889120591
le1
Γ (120574)21minus120583 max
119909120591isin[minus11]
100381610038161003816100381610038161(119909 120591)
10038161003816100381610038161003816int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
≜ 1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
(70)
letting 120578 = 119902120591 + 119902 minus 1 119911 = 119902120579 + 119902 minus 1 we have
1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
119890120574
(119911) 119889119911 119889120578
=1
Γ (120574)119902int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
119902120591+119902minus1
minus1
(119909 minus 119911)minus120583
119890120574
(119911) 119889119911) 119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
120591
minus1
(1 minus 120579)minus120583
119890120574
(119902120579 + 119902 minus 1) 119889120579)119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
(int
119909
120579
1198702(119909 119902120591 + 119902 minus 1)
times (1 minus 120579)
minus120583
119889120591) 119890120574
times (119902120579 + 119902 minus 1) 119889120579
le1
Γ (120574)21minus120583
1199021+120574 max
119909120591isin[minus11]
10038161003816100381610038161198702(119909 120591)
1003816100381610038161003816
times int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
≜ 1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
(71)
Then (67) gives
119890120574
(119909) le1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
+1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
+ 119867 (119909) + 1198691(119909) + 119869
2(119909) + 119869
3(119909)
+ 1198694(119909) + 119869
5(119909) + 119869
6(119909) + 119869
7(119909)
(72)
It follows from the Gronwall inequality that
1003817100381710038171003817119890120574
(119909)1003817100381710038171003817119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(73)
From (65) we have
119890 (119909)119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(74)
Advances in Mathematical Physics 9
We now apply Lemma 7 to obtain that
100381710038171003817100381711986911003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
100381710038171003817100381711986921003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
(75)
Due to Lemma 2
100381710038171003817100381711986931003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus1198981003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log1198731003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986941003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(76)
By virtue of Lemma 2 (33b) with119898 = 1
100381710038171003817100381711986961003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
100381710038171003817100381711986971003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
(77)
We now estimate the term 1198698(119909) It follows from Lemmas
5 and 6 with 119870(119909 120591) = 1Γ(120574) that
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
12057410038171003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873M119890
120574
)10038171003817100381710038171003817119871infin(119868)
le (1 +10038171003817100381710038171003817119868minus120583minus120583
119873
10038171003817100381710038171003817119871infin(119868))119862119873
minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581
le
119862119873(12)minus120583minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus120581 log1198731003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
1
2le 120583 lt 1
(78)
where in the last step we have used Lemma 6 under thefollowing assumption
1
2minus 120583 lt 120581 lt 1 minus 120583 when 0 lt 120583 lt
1
2
0 lt 120581 lt 1 minus 120583 when 1
2le 120583 lt 1
(79)
Provided that119873 is sufficiently large combining (75) (76)(77) and (78) gives1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)1003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
119880 (119909) minus 119906 (119909)119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
(80)
Using 120574 = 1 minus 120583 we have the desired estimates (46) and (47)
Next we will give the error estimates in 1198712
120596minus120583minus120583 space
Theorem 10 If the hypotheses given in Theorem 9 hold then1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)10038171003817100381710038171198712120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
119880(119909) minus 119906(119909)1198712
120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
(81)
10 Advances in Mathematical Physics
for any 120581 isin (0 120574) provided that119873 is sufficiently large and 119862 isa constant independent of119873 where
1198811= 119870
lowast
(1199061198712(119868)
+1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868))
1198812= 119870
lowast
1199061198712(119868)
(82)
Proof Byusing the generalization ofGronwalls Lemma 4 andthe Hardy inequality Lemma 8 it follows from (72) that
100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
1198901198712
120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
(83)
Now using Lemma 7 we have10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
1(119906
1198712(119868)
+ 119890119871infin(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
2(119906
1198712(119868)
+ 119890119871infin(119868))
(84)
By the convergence result in Theorem 9 (119898 = 1) we have
119890119871infin(119868)
le 119862(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868)) (85)
So that10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
1(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
2(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
(86)
Due to Lemma 2 (33a)10038171003817100381710038171198693
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1198981003816100381610038161003816119863120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
(87)
By virtue of Lemma 2 (33a) with119898 = 1
1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
10038161003816100381610038161003816100381610038161003816
int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
100381610038161003816100381610038161003816100381610038161003816
int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
(88)
Finally it follows from Lemmas 5 and 7 that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
120574100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le10038171003817100381710038171003817119868minus120583minus120583
119873(M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
+1003817100381710038171003817M119890
120574
minusT11987311989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)
le 1198621003817100381710038171003817M119890
120574
minusT1198731198901205741003817100381710038171003817119871infin(119868)
le 119862119873minus1205811003817100381710038171003817M119890
12057410038171003817100381710038170120581
le 119862119873minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
(89)
where in the last step we used Lemma 6 for any 120581 isin (0 1minus120583)By the convergence result in Theorem 9 we obtain that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)
le
11986211987312minus120583minus119898minus120581
(119870lowast
1199061198712(119868)
+ 11987312U) 0 lt 120583 lt
1
2
119862119873minus119898minus120581 log119873(119870
lowast
1199061198712(119868)
+ 11987312U)
1
2le 120583 lt 1
(90)
for119873 sufficiently large and for any 120581 isin (0 1 minus 120583) The desiredestimates (81) follows from the previous estimates and (83)with 120574 = 1 minus 120583
5 Algorithm Implementation andNumerical Experiments
In this subsection we present the numerical results obtainedby the proposed collocation spectral method The estimatesin Section 4 indicates that the convergence of numericalsolutions is exponential if the exact solution is smooth Toconfirm the theoretical prediction a numerical experimentis carried out by considering the following example
Example 1 Consider the following fractional integrodifferen-tial equation
119863075
119910 (119905) =6119905225
Γ (325)minus1199052
119890119905
5119910 (119905) minus
(119902119905)2
5119910 (119902119905)
+ int
119905
0
119890119905
119904119910 (119904) 119889119904 + int
119902119905
0
120591119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 0
(91)
The corresponding exact solution is given by 119910(119905) = 1199053
Figure 1 presents the approximate and exact solutionson the left-hand side and presents the approximate and
Advances in Mathematical Physics 11
0 02 04 06 08 10
02
04
06
08
1
12
14
Exact solutionApproximate solution
(a)
0 02 04 06 08 10
05
1
15
2
25
Exact solutionApproximate solution
(b)
Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075
119910(119905) (b)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(a)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(b)
Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075
119910(119905) (b) versusthe number of collocation points in 119871
infin and 1198712
120596norms
exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871
infin and 1198712
120596minus120583minus120583 norms As
expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions
Example 2 Consider the following fractional integrodiffer-ential equation
119863120572
119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int
119905
0
119890119904(119905minus119904)
119910 (119904) 119889119904
+ int
119902119905
0
119902119905119890120591(119902119905minus120591)
119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 1
(92)
when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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6 Advances in Mathematical Physics
holds if and only if
sup119886lt119909lt119887
(int
119887
119909
119906 (119905) 119889119905)
1119902
(int
119909
119886
V1minus1199011015840
(119905) 119889119905)
11199011015840
lt infin 1199011015840
=119901
119901 minus 1
(44)
for the case 1 lt 119901 le 119902 lt infin Here 119879 is an operator of the form
(119879119865) (119909) = int
119909
119886
119896 (119909 119905) 119891 (119905) 119889119905 (45)
with 119896(119909 119905) a given kernel 119906 V are nonnegative weightfunctions and minusinfin le 119886 lt 119887 le infin
4 Convergence Analysis
This section is devoted to provide a convergence analysis forthe numerical scheme The goal is to show that the rate ofconvergence is exponential that is the spectral accuracy canbe obtained for the proposed approximations Firstly we willcarry our convergence analysis in 119871
infin space
Theorem 9 Let 119906(119909) be the exact solution of the fractionaldelay-integrodifferential equation (7)-(8) which is assumedto be sufficiently smooth Assume that 119880(119909) and 119880
120574
(119909) areobtained by using the spectral collocation scheme (26)-(27)together with a polynomial interpolation (25) If 120574 associatedwith the weakly singular kernel satisfies 0 lt 120574 lt 1 and119906 isin 119867
119898+1
120596minus120583minus120583(119868) then
1003817100381710038171003817119880120574
minus 119863120574
1199061003817100381710038171003817119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(46)
119880 minus 119906119871infin(119868)
le
119862119873120574minus12minus119898
(119870lowast
1199061198712(119868)
+ 11987312U)
1
2lt 120574 lt 1
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 11987312U) 0 lt 120574 le
1
2
(47)
provided that 119873 is sufficiently large where 119862 is a constantindependent of 119873 but which will depend on the bounds of thefunctions 119870(119909 119904) and the index 120583
119870lowast
= max119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
+ max119909isin[minus11]
10038161003816100381610038161198702(119909 120578 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(48)
U =1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868) (49)
Proof We let
(119870 (119904 (119909119894 ])) 120601 (119904 (119909
119894 ])))
119873
=
119873
sum
119896=0
119870(119904 (119909119894 ]119896)) 120601 (119904 (119909
119894 ]119896)) 120596
00
119896
(50)
The numerical scheme (26)-(27) can be written as
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2(119870
1(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
119880 (120579 (119909minus120583minus120583
119894 ])))
119873
+
119902 (1 + 119909minus120583minus120583
119894)
2(119870
2(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 ]))
119880 (120578 (119909minus120583minus120583
119894 ])))
119873
(51)
119906119894=
1
Γ (120574)(119879
2)
120574
int
119909119894
minus1
(119909119894minus 120579)
minus120583
119880120574
(120579) 119889120579 + 119906minus1 (52)
which gives
119906120574
119894= 119906
119894+ (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+1 + 119909
minus120583minus120583
119894
2int
1
minus1
1198701(119909
minus120583minus120583
119894 120579 (119909
minus120583minus120583
119894 ]))
times 119880 (120579 (119909minus120583minus120583
119894 ])) 119889] + 119868
1(119909
minus120583minus120583
119894)
+
119902 (1 + 119909minus120583minus120583
119894)
2int
1
minus1
1198702(119909
minus120583minus120583
119894 120578 (119909
minus120583minus120583
119894 120585))
times 119880 (120578 (119909minus120583minus120583
119894 120585)) 119889120585 + 119868
2(119909
minus120583minus120583
119894)
(53)
where
1198681(119909) =
1 + 119909
2int
1
minus1
1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]
minus1 + 119909
2(119870
1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))
119873
1198682(119909) =
119902 (1 + 119909)
2int
1
minus1
1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585
minus119902 (1 + 119909)
2(119870
2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))
119873
(54)
Advances in Mathematical Physics 7
Using the integration error estimates from Jacobi-Gausspolynomials quadrature in Lemma 1 we have
10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
119880 (120579 (119909 sdot))1198712(119868)
le 119862119873minus119898
119870lowast
1119880
1198712(119868)
(55)
10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)
le 119862119873minus119898
119870lowast
2119880
1198712(119868)
(56)
119870lowast
1= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
119870lowast
2= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(57)
From (18) (53) can be rewritten as
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579)119880 (120579) 119889120579 + 119868
1(119909
minus120583minus120583
119894)
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578)119880 (120578) 119889120578
+ 1198682(119909
minus120583minus120583
119894)
(58)
Let 119890 and 119890120574 denote the error function
119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574
(119909) = 119880120574
(119909) minus 119906120574
(119909)
(59)
we have
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119906 (120579) 119889120579
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119890 (120579) 119889120579
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119906 (120578) 119889120578
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119890 (120578) 119889120578
+ 1198681(119909
minus120583minus120583
119894) + 119868
2(119909
minus120583minus120583
119894)
(60)
Multiplying119865119894(119909)onboth sides of (62) and (52) summing
up from 0 to119873 and using (7)-(8) yield
119880120574
(119909) = 119880 (119909) + 119868minus120583minus120583
119873(
119873
sum
119895=0
119906119895119865119895(119902119909 + 119902 minus 1))
+ 119868minus120583minus120583
119873119892 (119909)
+ 119868minus120583minus120583
119873[119863
120574
119906 (119909) minus 119906 (119909)
minus119906 (119902 (119909) + 119902 minus 1) minus 119892 (119909)]
+ 119868minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ 119868minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909)
(61)
119880 (119909) = 119868minus120583minus120583
119873(119906 (119909) minus 119906
minus1)
+ 119868minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
+ 119868minus120583minus120583
119873(119906
minus1)
(62)
where
1198691(119909) = 119868
minus120583minus120583
1198731198681(119909) 119869
2(119909) = 119868
minus120583minus120583
1198731198682(119909) (63)
From (61)-(62) we have
119890120574
(119909) = 119890 (119909) + int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(64)
119890 (119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+ 1198694(119909) + 119869
8(119909)
(65)
where
1198693(119909) = 119868
minus120583minus120583
119873119863120574
119906 (119909) minus 119863120574
119906 (119909)
1198694(119909) = 119868
minus120583minus120583
119873119906 (119909) minus 119906 (119909)
1198695(119909) = 119868
minus120583minus120583
119873119906 (119902119909 + 119902 minus 1) minus 119906 (119902119909 + 119902 minus 1)
1198696(119909) = 119868
minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
minus int
119909
minus1
119870 (119909 119904) 119890 (119904) 119889119904
8 Advances in Mathematical Physics
1198697(119909) = 119868
minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
minus int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1198698(119909) = 119868
minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
minus1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
(66)
Due to (64) we obtain
119890120574
(119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
times 119890120574
(119911) 119889119911 119889120578 + 119867 (119909)
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(67)
where
119867(119909) = int
119909
minus1
1198701(119909 120579) int
120579
minus1
(1198694(120579) + 119869
8(120579)) 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(1198694(120578) + 119869
8(120578)) 119889120578
+ 1198694(119909) + 119869
8(119909)
(68)
Using the Dirichletrsquos formula which sates that
int
119909
minus1
int
120579
minus1
Φ (120579 120591) 119889120591 119889120579 = int
119909
minus1
int
119909
120591
Φ (120579 120591) 119889120579 119889120591 (69)
provided that the integral exists we obtain
1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
=1
Γ (120574)int
119909
minus1
(int
119909
120579
1198701(119909 120579) (120579 minus 120591)
minus120583
119889120579) 119890120574
(120591) 119889120591
le1
Γ (120574)21minus120583 max
119909120591isin[minus11]
100381610038161003816100381610038161(119909 120591)
10038161003816100381610038161003816int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
≜ 1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
(70)
letting 120578 = 119902120591 + 119902 minus 1 119911 = 119902120579 + 119902 minus 1 we have
1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
119890120574
(119911) 119889119911 119889120578
=1
Γ (120574)119902int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
119902120591+119902minus1
minus1
(119909 minus 119911)minus120583
119890120574
(119911) 119889119911) 119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
120591
minus1
(1 minus 120579)minus120583
119890120574
(119902120579 + 119902 minus 1) 119889120579)119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
(int
119909
120579
1198702(119909 119902120591 + 119902 minus 1)
times (1 minus 120579)
minus120583
119889120591) 119890120574
times (119902120579 + 119902 minus 1) 119889120579
le1
Γ (120574)21minus120583
1199021+120574 max
119909120591isin[minus11]
10038161003816100381610038161198702(119909 120591)
1003816100381610038161003816
times int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
≜ 1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
(71)
Then (67) gives
119890120574
(119909) le1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
+1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
+ 119867 (119909) + 1198691(119909) + 119869
2(119909) + 119869
3(119909)
+ 1198694(119909) + 119869
5(119909) + 119869
6(119909) + 119869
7(119909)
(72)
It follows from the Gronwall inequality that
1003817100381710038171003817119890120574
(119909)1003817100381710038171003817119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(73)
From (65) we have
119890 (119909)119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(74)
Advances in Mathematical Physics 9
We now apply Lemma 7 to obtain that
100381710038171003817100381711986911003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
100381710038171003817100381711986921003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
(75)
Due to Lemma 2
100381710038171003817100381711986931003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus1198981003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log1198731003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986941003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(76)
By virtue of Lemma 2 (33b) with119898 = 1
100381710038171003817100381711986961003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
100381710038171003817100381711986971003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
(77)
We now estimate the term 1198698(119909) It follows from Lemmas
5 and 6 with 119870(119909 120591) = 1Γ(120574) that
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
12057410038171003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873M119890
120574
)10038171003817100381710038171003817119871infin(119868)
le (1 +10038171003817100381710038171003817119868minus120583minus120583
119873
10038171003817100381710038171003817119871infin(119868))119862119873
minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581
le
119862119873(12)minus120583minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus120581 log1198731003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
1
2le 120583 lt 1
(78)
where in the last step we have used Lemma 6 under thefollowing assumption
1
2minus 120583 lt 120581 lt 1 minus 120583 when 0 lt 120583 lt
1
2
0 lt 120581 lt 1 minus 120583 when 1
2le 120583 lt 1
(79)
Provided that119873 is sufficiently large combining (75) (76)(77) and (78) gives1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)1003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
119880 (119909) minus 119906 (119909)119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
(80)
Using 120574 = 1 minus 120583 we have the desired estimates (46) and (47)
Next we will give the error estimates in 1198712
120596minus120583minus120583 space
Theorem 10 If the hypotheses given in Theorem 9 hold then1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)10038171003817100381710038171198712120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
119880(119909) minus 119906(119909)1198712
120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
(81)
10 Advances in Mathematical Physics
for any 120581 isin (0 120574) provided that119873 is sufficiently large and 119862 isa constant independent of119873 where
1198811= 119870
lowast
(1199061198712(119868)
+1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868))
1198812= 119870
lowast
1199061198712(119868)
(82)
Proof Byusing the generalization ofGronwalls Lemma 4 andthe Hardy inequality Lemma 8 it follows from (72) that
100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
1198901198712
120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
(83)
Now using Lemma 7 we have10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
1(119906
1198712(119868)
+ 119890119871infin(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
2(119906
1198712(119868)
+ 119890119871infin(119868))
(84)
By the convergence result in Theorem 9 (119898 = 1) we have
119890119871infin(119868)
le 119862(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868)) (85)
So that10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
1(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
2(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
(86)
Due to Lemma 2 (33a)10038171003817100381710038171198693
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1198981003816100381610038161003816119863120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
(87)
By virtue of Lemma 2 (33a) with119898 = 1
1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
10038161003816100381610038161003816100381610038161003816
int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
100381610038161003816100381610038161003816100381610038161003816
int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
(88)
Finally it follows from Lemmas 5 and 7 that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
120574100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le10038171003817100381710038171003817119868minus120583minus120583
119873(M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
+1003817100381710038171003817M119890
120574
minusT11987311989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)
le 1198621003817100381710038171003817M119890
120574
minusT1198731198901205741003817100381710038171003817119871infin(119868)
le 119862119873minus1205811003817100381710038171003817M119890
12057410038171003817100381710038170120581
le 119862119873minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
(89)
where in the last step we used Lemma 6 for any 120581 isin (0 1minus120583)By the convergence result in Theorem 9 we obtain that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)
le
11986211987312minus120583minus119898minus120581
(119870lowast
1199061198712(119868)
+ 11987312U) 0 lt 120583 lt
1
2
119862119873minus119898minus120581 log119873(119870
lowast
1199061198712(119868)
+ 11987312U)
1
2le 120583 lt 1
(90)
for119873 sufficiently large and for any 120581 isin (0 1 minus 120583) The desiredestimates (81) follows from the previous estimates and (83)with 120574 = 1 minus 120583
5 Algorithm Implementation andNumerical Experiments
In this subsection we present the numerical results obtainedby the proposed collocation spectral method The estimatesin Section 4 indicates that the convergence of numericalsolutions is exponential if the exact solution is smooth Toconfirm the theoretical prediction a numerical experimentis carried out by considering the following example
Example 1 Consider the following fractional integrodifferen-tial equation
119863075
119910 (119905) =6119905225
Γ (325)minus1199052
119890119905
5119910 (119905) minus
(119902119905)2
5119910 (119902119905)
+ int
119905
0
119890119905
119904119910 (119904) 119889119904 + int
119902119905
0
120591119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 0
(91)
The corresponding exact solution is given by 119910(119905) = 1199053
Figure 1 presents the approximate and exact solutionson the left-hand side and presents the approximate and
Advances in Mathematical Physics 11
0 02 04 06 08 10
02
04
06
08
1
12
14
Exact solutionApproximate solution
(a)
0 02 04 06 08 10
05
1
15
2
25
Exact solutionApproximate solution
(b)
Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075
119910(119905) (b)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(a)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(b)
Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075
119910(119905) (b) versusthe number of collocation points in 119871
infin and 1198712
120596norms
exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871
infin and 1198712
120596minus120583minus120583 norms As
expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions
Example 2 Consider the following fractional integrodiffer-ential equation
119863120572
119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int
119905
0
119890119904(119905minus119904)
119910 (119904) 119889119904
+ int
119902119905
0
119902119905119890120591(119902119905minus120591)
119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 1
(92)
when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Advances in Mathematical Physics 7
Using the integration error estimates from Jacobi-Gausspolynomials quadrature in Lemma 1 we have
10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
119880 (120579 (119909 sdot))1198712(119868)
le 119862119873minus119898
119870lowast
1119880
1198712(119868)
(55)
10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873
minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))
1003816100381610038161003816100381611986711989811987312059600(119868)
1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)
le 119862119873minus119898
119870lowast
2119880
1198712(119868)
(56)
119870lowast
1= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
119870lowast
2= max
119909isin[minus11]
10038161003816100381610038161198701(119909 120579 (119909 sdot))
100381610038161003816100381611989811987312059600(119868)
(57)
From (18) (53) can be rewritten as
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579)119880 (120579) 119889120579 + 119868
1(119909
minus120583minus120583
119894)
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578)119880 (120578) 119889120578
+ 1198682(119909
minus120583minus120583
119894)
(58)
Let 119890 and 119890120574 denote the error function
119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574
(119909) = 119880120574
(119909) minus 119906120574
(119909)
(59)
we have
119906120574
119894= 119886 (119909
minus120583minus120583
119894) 119906
119894+ 119887 (119909
minus120583minus120583
119894)
times (
119873
sum
119895=0
119906119895119865119895(119902119909
minus120583minus120583
119894+ 119902 minus 1)) + 119892 (119909
minus120583minus120583
119894)
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119906 (120579) 119889120579
+ int
119909minus120583minus120583
119894
minus1
1198701(119909
minus120583minus120583
119894 120579) 119890 (120579) 119889120579
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119906 (120578) 119889120578
+ int
119902119909minus120583minus120583
119894+119902minus1
minus1
1198702(119909
minus120583minus120583
119894 120578) 119890 (120578) 119889120578
+ 1198681(119909
minus120583minus120583
119894) + 119868
2(119909
minus120583minus120583
119894)
(60)
Multiplying119865119894(119909)onboth sides of (62) and (52) summing
up from 0 to119873 and using (7)-(8) yield
119880120574
(119909) = 119880 (119909) + 119868minus120583minus120583
119873(
119873
sum
119895=0
119906119895119865119895(119902119909 + 119902 minus 1))
+ 119868minus120583minus120583
119873119892 (119909)
+ 119868minus120583minus120583
119873[119863
120574
119906 (119909) minus 119906 (119909)
minus119906 (119902 (119909) + 119902 minus 1) minus 119892 (119909)]
+ 119868minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ 119868minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909)
(61)
119880 (119909) = 119868minus120583minus120583
119873(119906 (119909) minus 119906
minus1)
+ 119868minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
+ 119868minus120583minus120583
119873(119906
minus1)
(62)
where
1198691(119909) = 119868
minus120583minus120583
1198731198681(119909) 119869
2(119909) = 119868
minus120583minus120583
1198731198682(119909) (63)
From (61)-(62) we have
119890120574
(119909) = 119890 (119909) + int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
+ int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(64)
119890 (119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+ 1198694(119909) + 119869
8(119909)
(65)
where
1198693(119909) = 119868
minus120583minus120583
119873119863120574
119906 (119909) minus 119863120574
119906 (119909)
1198694(119909) = 119868
minus120583minus120583
119873119906 (119909) minus 119906 (119909)
1198695(119909) = 119868
minus120583minus120583
119873119906 (119902119909 + 119902 minus 1) minus 119906 (119902119909 + 119902 minus 1)
1198696(119909) = 119868
minus120583minus120583
119873int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
minus int
119909
minus1
119870 (119909 119904) 119890 (119904) 119889119904
8 Advances in Mathematical Physics
1198697(119909) = 119868
minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
minus int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1198698(119909) = 119868
minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
minus1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
(66)
Due to (64) we obtain
119890120574
(119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
times 119890120574
(119911) 119889119911 119889120578 + 119867 (119909)
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(67)
where
119867(119909) = int
119909
minus1
1198701(119909 120579) int
120579
minus1
(1198694(120579) + 119869
8(120579)) 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(1198694(120578) + 119869
8(120578)) 119889120578
+ 1198694(119909) + 119869
8(119909)
(68)
Using the Dirichletrsquos formula which sates that
int
119909
minus1
int
120579
minus1
Φ (120579 120591) 119889120591 119889120579 = int
119909
minus1
int
119909
120591
Φ (120579 120591) 119889120579 119889120591 (69)
provided that the integral exists we obtain
1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
=1
Γ (120574)int
119909
minus1
(int
119909
120579
1198701(119909 120579) (120579 minus 120591)
minus120583
119889120579) 119890120574
(120591) 119889120591
le1
Γ (120574)21minus120583 max
119909120591isin[minus11]
100381610038161003816100381610038161(119909 120591)
10038161003816100381610038161003816int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
≜ 1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
(70)
letting 120578 = 119902120591 + 119902 minus 1 119911 = 119902120579 + 119902 minus 1 we have
1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
119890120574
(119911) 119889119911 119889120578
=1
Γ (120574)119902int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
119902120591+119902minus1
minus1
(119909 minus 119911)minus120583
119890120574
(119911) 119889119911) 119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
120591
minus1
(1 minus 120579)minus120583
119890120574
(119902120579 + 119902 minus 1) 119889120579)119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
(int
119909
120579
1198702(119909 119902120591 + 119902 minus 1)
times (1 minus 120579)
minus120583
119889120591) 119890120574
times (119902120579 + 119902 minus 1) 119889120579
le1
Γ (120574)21minus120583
1199021+120574 max
119909120591isin[minus11]
10038161003816100381610038161198702(119909 120591)
1003816100381610038161003816
times int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
≜ 1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
(71)
Then (67) gives
119890120574
(119909) le1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
+1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
+ 119867 (119909) + 1198691(119909) + 119869
2(119909) + 119869
3(119909)
+ 1198694(119909) + 119869
5(119909) + 119869
6(119909) + 119869
7(119909)
(72)
It follows from the Gronwall inequality that
1003817100381710038171003817119890120574
(119909)1003817100381710038171003817119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(73)
From (65) we have
119890 (119909)119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(74)
Advances in Mathematical Physics 9
We now apply Lemma 7 to obtain that
100381710038171003817100381711986911003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
100381710038171003817100381711986921003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
(75)
Due to Lemma 2
100381710038171003817100381711986931003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus1198981003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log1198731003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986941003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(76)
By virtue of Lemma 2 (33b) with119898 = 1
100381710038171003817100381711986961003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
100381710038171003817100381711986971003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
(77)
We now estimate the term 1198698(119909) It follows from Lemmas
5 and 6 with 119870(119909 120591) = 1Γ(120574) that
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
12057410038171003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873M119890
120574
)10038171003817100381710038171003817119871infin(119868)
le (1 +10038171003817100381710038171003817119868minus120583minus120583
119873
10038171003817100381710038171003817119871infin(119868))119862119873
minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581
le
119862119873(12)minus120583minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus120581 log1198731003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
1
2le 120583 lt 1
(78)
where in the last step we have used Lemma 6 under thefollowing assumption
1
2minus 120583 lt 120581 lt 1 minus 120583 when 0 lt 120583 lt
1
2
0 lt 120581 lt 1 minus 120583 when 1
2le 120583 lt 1
(79)
Provided that119873 is sufficiently large combining (75) (76)(77) and (78) gives1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)1003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
119880 (119909) minus 119906 (119909)119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
(80)
Using 120574 = 1 minus 120583 we have the desired estimates (46) and (47)
Next we will give the error estimates in 1198712
120596minus120583minus120583 space
Theorem 10 If the hypotheses given in Theorem 9 hold then1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)10038171003817100381710038171198712120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
119880(119909) minus 119906(119909)1198712
120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
(81)
10 Advances in Mathematical Physics
for any 120581 isin (0 120574) provided that119873 is sufficiently large and 119862 isa constant independent of119873 where
1198811= 119870
lowast
(1199061198712(119868)
+1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868))
1198812= 119870
lowast
1199061198712(119868)
(82)
Proof Byusing the generalization ofGronwalls Lemma 4 andthe Hardy inequality Lemma 8 it follows from (72) that
100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
1198901198712
120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
(83)
Now using Lemma 7 we have10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
1(119906
1198712(119868)
+ 119890119871infin(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
2(119906
1198712(119868)
+ 119890119871infin(119868))
(84)
By the convergence result in Theorem 9 (119898 = 1) we have
119890119871infin(119868)
le 119862(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868)) (85)
So that10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
1(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
2(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
(86)
Due to Lemma 2 (33a)10038171003817100381710038171198693
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1198981003816100381610038161003816119863120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
(87)
By virtue of Lemma 2 (33a) with119898 = 1
1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
10038161003816100381610038161003816100381610038161003816
int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
100381610038161003816100381610038161003816100381610038161003816
int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
(88)
Finally it follows from Lemmas 5 and 7 that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
120574100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le10038171003817100381710038171003817119868minus120583minus120583
119873(M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
+1003817100381710038171003817M119890
120574
minusT11987311989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)
le 1198621003817100381710038171003817M119890
120574
minusT1198731198901205741003817100381710038171003817119871infin(119868)
le 119862119873minus1205811003817100381710038171003817M119890
12057410038171003817100381710038170120581
le 119862119873minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
(89)
where in the last step we used Lemma 6 for any 120581 isin (0 1minus120583)By the convergence result in Theorem 9 we obtain that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)
le
11986211987312minus120583minus119898minus120581
(119870lowast
1199061198712(119868)
+ 11987312U) 0 lt 120583 lt
1
2
119862119873minus119898minus120581 log119873(119870
lowast
1199061198712(119868)
+ 11987312U)
1
2le 120583 lt 1
(90)
for119873 sufficiently large and for any 120581 isin (0 1 minus 120583) The desiredestimates (81) follows from the previous estimates and (83)with 120574 = 1 minus 120583
5 Algorithm Implementation andNumerical Experiments
In this subsection we present the numerical results obtainedby the proposed collocation spectral method The estimatesin Section 4 indicates that the convergence of numericalsolutions is exponential if the exact solution is smooth Toconfirm the theoretical prediction a numerical experimentis carried out by considering the following example
Example 1 Consider the following fractional integrodifferen-tial equation
119863075
119910 (119905) =6119905225
Γ (325)minus1199052
119890119905
5119910 (119905) minus
(119902119905)2
5119910 (119902119905)
+ int
119905
0
119890119905
119904119910 (119904) 119889119904 + int
119902119905
0
120591119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 0
(91)
The corresponding exact solution is given by 119910(119905) = 1199053
Figure 1 presents the approximate and exact solutionson the left-hand side and presents the approximate and
Advances in Mathematical Physics 11
0 02 04 06 08 10
02
04
06
08
1
12
14
Exact solutionApproximate solution
(a)
0 02 04 06 08 10
05
1
15
2
25
Exact solutionApproximate solution
(b)
Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075
119910(119905) (b)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(a)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(b)
Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075
119910(119905) (b) versusthe number of collocation points in 119871
infin and 1198712
120596norms
exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871
infin and 1198712
120596minus120583minus120583 norms As
expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions
Example 2 Consider the following fractional integrodiffer-ential equation
119863120572
119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int
119905
0
119890119904(119905minus119904)
119910 (119904) 119889119904
+ int
119902119905
0
119902119905119890120591(119902119905minus120591)
119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 1
(92)
when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Advances in Mathematical Physics
1198697(119909) = 119868
minus120583minus120583
119873int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
minus int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1198698(119909) = 119868
minus120583minus120583
119873(
1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579)
minus1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
(66)
Due to (64) we obtain
119890120574
(119909) =1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
times 119890120574
(119911) 119889119911 119889120578 + 119867 (119909)
+ 1198691(119909) + 119869
2(119909) + 119869
3(119909) + 119869
4(119909)
+ 1198695(119909) + 119869
6(119909) + 119869
7(119909)
(67)
where
119867(119909) = int
119909
minus1
1198701(119909 120579) int
120579
minus1
(1198694(120579) + 119869
8(120579)) 119889120579
+1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(1198694(120578) + 119869
8(120578)) 119889120578
+ 1198694(119909) + 119869
8(119909)
(68)
Using the Dirichletrsquos formula which sates that
int
119909
minus1
int
120579
minus1
Φ (120579 120591) 119889120591 119889120579 = int
119909
minus1
int
119909
120591
Φ (120579 120591) 119889120579 119889120591 (69)
provided that the integral exists we obtain
1
Γ (120574)int
119909
minus1
1198701(119909 120579) int
120579
minus1
(120579 minus 120591)minus120583
119890120574
(120591) 119889120591 119889120579
=1
Γ (120574)int
119909
minus1
(int
119909
120579
1198701(119909 120579) (120579 minus 120591)
minus120583
119889120579) 119890120574
(120591) 119889120591
le1
Γ (120574)21minus120583 max
119909120591isin[minus11]
100381610038161003816100381610038161(119909 120591)
10038161003816100381610038161003816int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
≜ 1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
(70)
letting 120578 = 119902120591 + 119902 minus 1 119911 = 119902120579 + 119902 minus 1 we have
1
Γ (120574)int
119902119909+119902minus1
minus1
1198702(119909 120578) int
120578
minus1
(120578 minus 119911)minus120583
119890120574
(119911) 119889119911 119889120578
=1
Γ (120574)119902int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
119902120591+119902minus1
minus1
(119909 minus 119911)minus120583
119890120574
(119911) 119889119911) 119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
1198702(119909 119902120591 + 119902 minus 1)
times (int
120591
minus1
(1 minus 120579)minus120583
119890120574
(119902120579 + 119902 minus 1) 119889120579)119889120591
=1
Γ (120574)1199021+120574
int
119909
minus1
(int
119909
120579
1198702(119909 119902120591 + 119902 minus 1)
times (1 minus 120579)
minus120583
119889120591) 119890120574
times (119902120579 + 119902 minus 1) 119889120579
le1
Γ (120574)21minus120583
1199021+120574 max
119909120591isin[minus11]
10038161003816100381610038161198702(119909 120591)
1003816100381610038161003816
times int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
≜ 1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
(71)
Then (67) gives
119890120574
(119909) le1
Γ (120574)int
119909
minus1
(119909 minus 120579)minus120583
119890120574
(120579) 119889120579
+1198721int
119909
minus1
1003816100381610038161003816119890120574
(120591)1003816100381610038161003816 119889120591
+1198722int
119909
minus1
1003816100381610038161003816119890120574
(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579
+ 119867 (119909) + 1198691(119909) + 119869
2(119909) + 119869
3(119909)
+ 1198694(119909) + 119869
5(119909) + 119869
6(119909) + 119869
7(119909)
(72)
It follows from the Gronwall inequality that
1003817100381710038171003817119890120574
(119909)1003817100381710038171003817119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(73)
From (65) we have
119890 (119909)119871infin(119868)
le
8
sum
119894=1
1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)
(74)
Advances in Mathematical Physics 9
We now apply Lemma 7 to obtain that
100381710038171003817100381711986911003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
100381710038171003817100381711986921003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
(75)
Due to Lemma 2
100381710038171003817100381711986931003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus1198981003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log1198731003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986941003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(76)
By virtue of Lemma 2 (33b) with119898 = 1
100381710038171003817100381711986961003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
100381710038171003817100381711986971003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
(77)
We now estimate the term 1198698(119909) It follows from Lemmas
5 and 6 with 119870(119909 120591) = 1Γ(120574) that
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
12057410038171003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873M119890
120574
)10038171003817100381710038171003817119871infin(119868)
le (1 +10038171003817100381710038171003817119868minus120583minus120583
119873
10038171003817100381710038171003817119871infin(119868))119862119873
minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581
le
119862119873(12)minus120583minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus120581 log1198731003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
1
2le 120583 lt 1
(78)
where in the last step we have used Lemma 6 under thefollowing assumption
1
2minus 120583 lt 120581 lt 1 minus 120583 when 0 lt 120583 lt
1
2
0 lt 120581 lt 1 minus 120583 when 1
2le 120583 lt 1
(79)
Provided that119873 is sufficiently large combining (75) (76)(77) and (78) gives1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)1003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
119880 (119909) minus 119906 (119909)119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
(80)
Using 120574 = 1 minus 120583 we have the desired estimates (46) and (47)
Next we will give the error estimates in 1198712
120596minus120583minus120583 space
Theorem 10 If the hypotheses given in Theorem 9 hold then1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)10038171003817100381710038171198712120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
119880(119909) minus 119906(119909)1198712
120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
(81)
10 Advances in Mathematical Physics
for any 120581 isin (0 120574) provided that119873 is sufficiently large and 119862 isa constant independent of119873 where
1198811= 119870
lowast
(1199061198712(119868)
+1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868))
1198812= 119870
lowast
1199061198712(119868)
(82)
Proof Byusing the generalization ofGronwalls Lemma 4 andthe Hardy inequality Lemma 8 it follows from (72) that
100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
1198901198712
120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
(83)
Now using Lemma 7 we have10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
1(119906
1198712(119868)
+ 119890119871infin(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
2(119906
1198712(119868)
+ 119890119871infin(119868))
(84)
By the convergence result in Theorem 9 (119898 = 1) we have
119890119871infin(119868)
le 119862(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868)) (85)
So that10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
1(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
2(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
(86)
Due to Lemma 2 (33a)10038171003817100381710038171198693
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1198981003816100381610038161003816119863120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
(87)
By virtue of Lemma 2 (33a) with119898 = 1
1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
10038161003816100381610038161003816100381610038161003816
int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
100381610038161003816100381610038161003816100381610038161003816
int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
(88)
Finally it follows from Lemmas 5 and 7 that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
120574100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le10038171003817100381710038171003817119868minus120583minus120583
119873(M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
+1003817100381710038171003817M119890
120574
minusT11987311989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)
le 1198621003817100381710038171003817M119890
120574
minusT1198731198901205741003817100381710038171003817119871infin(119868)
le 119862119873minus1205811003817100381710038171003817M119890
12057410038171003817100381710038170120581
le 119862119873minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
(89)
where in the last step we used Lemma 6 for any 120581 isin (0 1minus120583)By the convergence result in Theorem 9 we obtain that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)
le
11986211987312minus120583minus119898minus120581
(119870lowast
1199061198712(119868)
+ 11987312U) 0 lt 120583 lt
1
2
119862119873minus119898minus120581 log119873(119870
lowast
1199061198712(119868)
+ 11987312U)
1
2le 120583 lt 1
(90)
for119873 sufficiently large and for any 120581 isin (0 1 minus 120583) The desiredestimates (81) follows from the previous estimates and (83)with 120574 = 1 minus 120583
5 Algorithm Implementation andNumerical Experiments
In this subsection we present the numerical results obtainedby the proposed collocation spectral method The estimatesin Section 4 indicates that the convergence of numericalsolutions is exponential if the exact solution is smooth Toconfirm the theoretical prediction a numerical experimentis carried out by considering the following example
Example 1 Consider the following fractional integrodifferen-tial equation
119863075
119910 (119905) =6119905225
Γ (325)minus1199052
119890119905
5119910 (119905) minus
(119902119905)2
5119910 (119902119905)
+ int
119905
0
119890119905
119904119910 (119904) 119889119904 + int
119902119905
0
120591119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 0
(91)
The corresponding exact solution is given by 119910(119905) = 1199053
Figure 1 presents the approximate and exact solutionson the left-hand side and presents the approximate and
Advances in Mathematical Physics 11
0 02 04 06 08 10
02
04
06
08
1
12
14
Exact solutionApproximate solution
(a)
0 02 04 06 08 10
05
1
15
2
25
Exact solutionApproximate solution
(b)
Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075
119910(119905) (b)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(a)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(b)
Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075
119910(119905) (b) versusthe number of collocation points in 119871
infin and 1198712
120596norms
exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871
infin and 1198712
120596minus120583minus120583 norms As
expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions
Example 2 Consider the following fractional integrodiffer-ential equation
119863120572
119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int
119905
0
119890119904(119905minus119904)
119910 (119904) 119889119904
+ int
119902119905
0
119902119905119890120591(119902119905minus120591)
119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 1
(92)
when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 9
We now apply Lemma 7 to obtain that
100381710038171003817100381711986911003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
1(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
100381710038171003817100381711986921003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583max
0le119894le119873
100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt
1
2
119862 log119873max0le119894le119873
100381610038161003816100381611986811989411003816100381610038161003816
1
2le 120583 lt 1
le
119862119873(12)minus120583minus119898
119870lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 0 lt 120583 lt1
2
119862119873minus119898 log119873119870
lowast
2(119906
1198712(119868)
+1003817100381710038171003817119890
120574
(119909)1003817100381710038171003817119871infin(119868)
) 1
2le 120583 lt 1
(75)
Due to Lemma 2
100381710038171003817100381711986931003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus1198981003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log1198731003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986941003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
le
1198621198731minus120583minus119898
|119906|119867119898119873
120596119888(119868) 0 lt 120583 lt
1
2
119862119873(12)minus119898 log119873|119906|
119867119898119873
120596119888(119868)
1
2le 120583 lt 1
(76)
By virtue of Lemma 2 (33b) with119898 = 1
100381710038171003817100381711986961003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
100381710038171003817100381711986971003817100381710038171003817119871infin(119868)
le
119862119873minus120583
119890119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus(12)
119890119871infin(119868)
1
2le 120583 lt 1
(77)
We now estimate the term 1198698(119909) It follows from Lemmas
5 and 6 with 119870(119909 120591) = 1Γ(120574) that
100381710038171003817100381711986951003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
12057410038171003817100381710038171003817119871infin(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873M119890
120574
)10038171003817100381710038171003817119871infin(119868)
le (1 +10038171003817100381710038171003817119868minus120583minus120583
119873
10038171003817100381710038171003817119871infin(119868))119862119873
minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581
le
119862119873(12)minus120583minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt
1
2
119862119873minus120581 log1198731003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
1
2le 120583 lt 1
(78)
where in the last step we have used Lemma 6 under thefollowing assumption
1
2minus 120583 lt 120581 lt 1 minus 120583 when 0 lt 120583 lt
1
2
0 lt 120581 lt 1 minus 120583 when 1
2le 120583 lt 1
(79)
Provided that119873 is sufficiently large combining (75) (76)(77) and (78) gives1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)1003817100381710038171003817119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
119880 (119909) minus 119906 (119909)119871infin(119868)
le
119862119873(12)minus120583minus119898
(119870lowast
1199061198712(119868)
+ 119873(12)U) 0 lt 120583 lt
1
2
119862119873minus119898 log119873(119870
lowast
1199061198712(119868)
+ 119873(12)U)
1
2le 120583 lt 1
(80)
Using 120574 = 1 minus 120583 we have the desired estimates (46) and (47)
Next we will give the error estimates in 1198712
120596minus120583minus120583 space
Theorem 10 If the hypotheses given in Theorem 9 hold then1003817100381710038171003817119880
120574
(119909) minus 119906120574
(119909)10038171003817100381710038171198712120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
119880(119909) minus 119906(119909)1198712
120596minus120583minus120583
(119868)
le
119862119873minus119898
(1198811+ 119873
120574minus(12)minus120581
1198812+ 119873
120574minus120581U) 1
2lt 120574 lt 1
119862119873minus119898
(1198811+ 119873
minus120581 log1198731198812+ 119873
(12)minus120581 log119873U) 0 lt 120574 le1
2
(81)
10 Advances in Mathematical Physics
for any 120581 isin (0 120574) provided that119873 is sufficiently large and 119862 isa constant independent of119873 where
1198811= 119870
lowast
(1199061198712(119868)
+1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868))
1198812= 119870
lowast
1199061198712(119868)
(82)
Proof Byusing the generalization ofGronwalls Lemma 4 andthe Hardy inequality Lemma 8 it follows from (72) that
100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
1198901198712
120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
(83)
Now using Lemma 7 we have10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
1(119906
1198712(119868)
+ 119890119871infin(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
2(119906
1198712(119868)
+ 119890119871infin(119868))
(84)
By the convergence result in Theorem 9 (119898 = 1) we have
119890119871infin(119868)
le 119862(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868)) (85)
So that10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
1(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
2(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
(86)
Due to Lemma 2 (33a)10038171003817100381710038171198693
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1198981003816100381610038161003816119863120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
(87)
By virtue of Lemma 2 (33a) with119898 = 1
1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
10038161003816100381610038161003816100381610038161003816
int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
100381610038161003816100381610038161003816100381610038161003816
int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
(88)
Finally it follows from Lemmas 5 and 7 that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
120574100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le10038171003817100381710038171003817119868minus120583minus120583
119873(M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
+1003817100381710038171003817M119890
120574
minusT11987311989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)
le 1198621003817100381710038171003817M119890
120574
minusT1198731198901205741003817100381710038171003817119871infin(119868)
le 119862119873minus1205811003817100381710038171003817M119890
12057410038171003817100381710038170120581
le 119862119873minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
(89)
where in the last step we used Lemma 6 for any 120581 isin (0 1minus120583)By the convergence result in Theorem 9 we obtain that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)
le
11986211987312minus120583minus119898minus120581
(119870lowast
1199061198712(119868)
+ 11987312U) 0 lt 120583 lt
1
2
119862119873minus119898minus120581 log119873(119870
lowast
1199061198712(119868)
+ 11987312U)
1
2le 120583 lt 1
(90)
for119873 sufficiently large and for any 120581 isin (0 1 minus 120583) The desiredestimates (81) follows from the previous estimates and (83)with 120574 = 1 minus 120583
5 Algorithm Implementation andNumerical Experiments
In this subsection we present the numerical results obtainedby the proposed collocation spectral method The estimatesin Section 4 indicates that the convergence of numericalsolutions is exponential if the exact solution is smooth Toconfirm the theoretical prediction a numerical experimentis carried out by considering the following example
Example 1 Consider the following fractional integrodifferen-tial equation
119863075
119910 (119905) =6119905225
Γ (325)minus1199052
119890119905
5119910 (119905) minus
(119902119905)2
5119910 (119902119905)
+ int
119905
0
119890119905
119904119910 (119904) 119889119904 + int
119902119905
0
120591119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 0
(91)
The corresponding exact solution is given by 119910(119905) = 1199053
Figure 1 presents the approximate and exact solutionson the left-hand side and presents the approximate and
Advances in Mathematical Physics 11
0 02 04 06 08 10
02
04
06
08
1
12
14
Exact solutionApproximate solution
(a)
0 02 04 06 08 10
05
1
15
2
25
Exact solutionApproximate solution
(b)
Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075
119910(119905) (b)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(a)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(b)
Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075
119910(119905) (b) versusthe number of collocation points in 119871
infin and 1198712
120596norms
exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871
infin and 1198712
120596minus120583minus120583 norms As
expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions
Example 2 Consider the following fractional integrodiffer-ential equation
119863120572
119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int
119905
0
119890119904(119905minus119904)
119910 (119904) 119889119904
+ int
119902119905
0
119902119905119890120591(119902119905minus120591)
119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 1
(92)
when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Advances in Mathematical Physics
for any 120581 isin (0 120574) provided that119873 is sufficiently large and 119862 isa constant independent of119873 where
1198811= 119870
lowast
(1199061198712(119868)
+1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868))
1198812= 119870
lowast
1199061198712(119868)
(82)
Proof Byusing the generalization ofGronwalls Lemma 4 andthe Hardy inequality Lemma 8 it follows from (72) that
100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
1198901198712
120596minus120583minus120583
(119868)le 119862
8
sum
119894=1
100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583
(119868)
(83)
Now using Lemma 7 we have10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
1(119906
1198712(119868)
+ 119890119871infin(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862 max
119909isin[minus11]
|119868 (119909)|
le 119862119873minus119898
119870lowast
2(119906
1198712(119868)
+ 119890119871infin(119868))
(84)
By the convergence result in Theorem 9 (119898 = 1) we have
119890119871infin(119868)
le 119862(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868)) (85)
So that10038171003817100381710038171198691
10038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
1(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583
(119868)
le 119862119873minus119898
119870lowast
2(1003816100381610038161003816119863
120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
+ |119906|119867119898119873
120596119888(119868)
+ 1199061198712(119868))
(86)
Due to Lemma 2 (33a)10038171003817100381710038171198693
10038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1198981003816100381610038161003816119863120574
1199061003816100381610038161003816119867119898119873120596119888(119868)
1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus119898
|119906|119867119898119873
120596119888(119868)
(87)
By virtue of Lemma 2 (33a) with119898 = 1
1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
10038161003816100381610038161003816100381610038161003816
int
119909
minus1
1198701(119909 120579) 119890 (120579) 119889120579
100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583
(119868)le 119862119873
minus1
100381610038161003816100381610038161003816100381610038161003816
int
119902119909+119902minus1
minus1
1198702(119909 120578) 119890 (120578) 119889120578
1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583
(119868)
le 119862119873minus1
1198901198712
120596minus120583minus120583
(119868)
(88)
Finally it follows from Lemmas 5 and 7 that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868)M119890
120574100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
=10038171003817100381710038171003817(119868
minus120583minus120583
119873minus 119868) (M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
le10038171003817100381710038171003817119868minus120583minus120583
119873(M119890
120574
minusT119873119890120574
)100381710038171003817100381710038171198712120596minus120583minus120583
(119868)
+1003817100381710038171003817M119890
120574
minusT11987311989012057410038171003817100381710038171198712120596minus120583minus120583
(119868)
le 1198621003817100381710038171003817M119890
120574
minusT1198731198901205741003817100381710038171003817119871infin(119868)
le 119862119873minus1205811003817100381710038171003817M119890
12057410038171003817100381710038170120581
le 119862119873minus1205811003817100381710038171003817119890
1205741003817100381710038171003817119871infin(119868)
(89)
where in the last step we used Lemma 6 for any 120581 isin (0 1minus120583)By the convergence result in Theorem 9 we obtain that
1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583
(119868)
le
11986211987312minus120583minus119898minus120581
(119870lowast
1199061198712(119868)
+ 11987312U) 0 lt 120583 lt
1
2
119862119873minus119898minus120581 log119873(119870
lowast
1199061198712(119868)
+ 11987312U)
1
2le 120583 lt 1
(90)
for119873 sufficiently large and for any 120581 isin (0 1 minus 120583) The desiredestimates (81) follows from the previous estimates and (83)with 120574 = 1 minus 120583
5 Algorithm Implementation andNumerical Experiments
In this subsection we present the numerical results obtainedby the proposed collocation spectral method The estimatesin Section 4 indicates that the convergence of numericalsolutions is exponential if the exact solution is smooth Toconfirm the theoretical prediction a numerical experimentis carried out by considering the following example
Example 1 Consider the following fractional integrodifferen-tial equation
119863075
119910 (119905) =6119905225
Γ (325)minus1199052
119890119905
5119910 (119905) minus
(119902119905)2
5119910 (119902119905)
+ int
119905
0
119890119905
119904119910 (119904) 119889119904 + int
119902119905
0
120591119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 0
(91)
The corresponding exact solution is given by 119910(119905) = 1199053
Figure 1 presents the approximate and exact solutionson the left-hand side and presents the approximate and
Advances in Mathematical Physics 11
0 02 04 06 08 10
02
04
06
08
1
12
14
Exact solutionApproximate solution
(a)
0 02 04 06 08 10
05
1
15
2
25
Exact solutionApproximate solution
(b)
Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075
119910(119905) (b)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(a)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(b)
Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075
119910(119905) (b) versusthe number of collocation points in 119871
infin and 1198712
120596norms
exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871
infin and 1198712
120596minus120583minus120583 norms As
expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions
Example 2 Consider the following fractional integrodiffer-ential equation
119863120572
119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int
119905
0
119890119904(119905minus119904)
119910 (119904) 119889119904
+ int
119902119905
0
119902119905119890120591(119902119905minus120591)
119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 1
(92)
when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 11
0 02 04 06 08 10
02
04
06
08
1
12
14
Exact solutionApproximate solution
(a)
0 02 04 06 08 10
05
1
15
2
25
Exact solutionApproximate solution
(b)
Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075
119910(119905) (b)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(a)
2 4 6 8 10 12 14 16
100
10minus5
10minus10
2 le N le 16
Linfin
L2
120596minus120583minus120583
-error-error
(b)
Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075
119910(119905) (b) versusthe number of collocation points in 119871
infin and 1198712
120596norms
exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871
infin and 1198712
120596minus120583minus120583 norms As
expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions
Example 2 Consider the following fractional integrodiffer-ential equation
119863120572
119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int
119905
0
119890119904(119905minus119904)
119910 (119904) 119889119904
+ int
119902119905
0
119902119905119890120591(119902119905minus120591)
119910 (120591) 119889120591 119905 isin [0 1]
119910 (0) = 1
(92)
when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Advances in Mathematical Physics
0 02 04 06 08 105
1
15
2
25
3
35
4
Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1
(a)
0 02 04 06 08 10
1
2
3
4
5
6
Approximate derivativeExact derivative
(b)
Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)
2 4 6 8 10 12 14 16 18 20
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 le N le 20
Linfin
L2
120596
(a)
10minus2
10minus4
10minus6
10minus8
10minus10
10minus12
10minus14
2 4 6 8 10 12 14 16 18 202 le N le 20
Linfin
L2
120596
(b)
Figure 4 Example 2 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 1199101015840
(119905) (b) versus thenumber of collocation points in 119871
infin and 1198712
120596norms
In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20
and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890
1199052
that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the
steps This figure shows the exponential rate of convergencepredicted by the proposed method
6 Conclusions and Future Work
This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 13
contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method
We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als
Acknowledgments
The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)
References
[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999
[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999
[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997
[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007
[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005
[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998
[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002
[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007
[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012
[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011
[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011
[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011
[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012
[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011
[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011
[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010
[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011
[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012
[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013
[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013
[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012
[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010
[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012
[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Advances in Mathematical Physics
[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001
[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989
[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970
[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971
[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998
[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984
[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003
[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of