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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 746401 12 pageshttpdxdoiorg1011552013746401
Research ArticleMultivariate Padeacute Approximation for Solving Nonlinear PartialDifferential Equations of Fractional Order
Veyis Turut1 and Nuran Guumlzel2
1 Department of Mathematics Faculty of Arts and Sciences Batman University Batman Turkey2Department of Mathematics Faculty of Arts and Sciences Yıldız Technical University Istanbul Turkey
Correspondence should be addressed to Nuran Guzel nguzelyildizedutr
Received 27 November 2012 Revised 14 January 2013 Accepted 14 January 2013
Academic Editor Hassan Eltayeb
Copyright copy 2013 V Turut and N Guzel 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
Two tecHniques were implemented the Adomian decompositionmethod (ADM) andmultivariate Pade approximation (MPA) forsolving nonlinear partial differential equations of fractional order The fractional derivatives are described in Caputo sense Firstthe fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM)then power series solution of fractional differential equation was put into multivariate Pade series Finally numerical results werecompared and presented in tables and figures
1 Introduction
Recently differential equations of fractional order havegained much interest in engineering physics chemistry andother sciences It can be said that the fractional derivative hasdrawn much attention due to its wide application in engi-neering physics [1ndash9] Some approximations and numericaltechniques have been used to provide an analytical approx-imation to linear and nonlinear differential equations andfractional differential equationsAmong them the variationaliteration method homotopy perturbation method [10 11]and the Adomian decomposition method are relatively newapproaches [5ndash9 12 13]
The decomposition method has been used to obtainapproximate solutions of a large class of linear or nonlineardifferential equations [12 13] Recently the application of themethod is extended for fractional differential equations [6ndash9 14]
Many definitions and theorems have been developed formultivariate Pade approximations MPA (see [15] for a sur-vey on multivariate Pade approximation) The multivariatePade Approximation has been used to obtain approximatesolutions of linear or nonlinear differential equations [16ndash19]
Recently the application of the unvariate Pade approximationis extended for fractional differential equations [20 21]
The objective of the present paper is to provide approxi-mate solutions for initial value problems of nonlinear partialdifferential equations of fractional order by usingmultivariatePade approximation
2 Definitions
For the concept of fractional derivative we will adoptCaputorsquos definition which is a modification of the Riemann-Liouville definition and has the advantage of dealing properlywith initial value problems in which the initial conditions aregiven in terms of the field variables and their integer orderwhich is the case in most physical processes The definitionscan be seen in [3 4 22 23]
3 Decomposition Method [24]
Consider
119863120572
lowast119905119906 (119909 119905) = 119891 (119906 119906
119909 119906119909119909) + 119892 (119909 119905) 119898 minus 1 lt 120572 le 119898
(1)
2 Abstract and Applied Analysis
The decomposition method requires that a nonlinear frac-tional differential equation (1) is expressed in terms of opera-tor form as
119863120572
lowast119905119906 (119909 119905) + 119871119906 (119909 119905) + 119873119906 (119909 119905) = 119892 (119909 119905) 119909 gt 0 (2)
where 119871 is a linear operator which might include other frac-tional derivatives of order less than 120572119873 is a nonlinear oper-ator which also might include other fractional derivatives oforder less than 120572 119863120572
lowast119905= 120597120572
120597119905120572 is the Caputo fractional
derivative of order 120572 and 119892(119909 119905) is the source function [24]Applying the operator 119869120572 [3 4 22 23] the inverse of the
operator 119863120572lowast119905 to both sides of (5) Odibat and Momani [24]
obtained
119906 (119909 119905) =
119898minus1
sum
119896=0
120597119896
119906
120597119905119896(119909 0+
)119905119896
119896+ 119869120572
119892 (119909 119905)
minus 119869120572
[119871119906 (119909 119905) + 119873119906 (119909 119905)]
infin
sum
119899=0
119906119899(119909 119905) =
119898minus1
sum
119896=0
120597119896
119906
120597119905119896(119909 0+
)119905119896
119896+ 119869120572
119892 (119909 119905)
minus 119869120572
[119871(
infin
sum
119899=0
119906119899(119909 119905)) +
infin
sum
119899=0
119860119899]
(3)
From this the iterates are determined in [24] by the followingrecursive way
1199060(119909 119905) =
119898minus1
sum
119896=0
120597119896
119906
120597119905119896(119909 0+
)119905119896
119896+ 119869120572
119892 (119909 119905)
1199061(119909 119905) = minus119869
120572
(1198711199060+ 1198600)
1199062(119909 119905) = minus119869
120572
(1198711199061+ 1198601)
119906119899+1(119909 119905) = minus119869
120572
(119871119906119899+ 119860119899)
(4)
4 Multivariate Padeacute Aproximation [25]
Consider the bivariate function 119891(119909 119910) with Taylor seriesdevelopment
119891 (119909 119910) =
infin
sum
119894119895=0
119888119894119895119909119894
119910119895 (5)
around the origin We know that a solution of unvariate Padeapproximation problem for
119891 (119909) =
infin
sum
119894=0
119888119894119909119894 (6)
is given by
119901 (119909) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119894=0
119888119894119909119894
119909
119898minus1
sum
119894=0
119888119894119909119894
119909119899
119898minus119899
sum
119894=0
119888119894119909119894
119888119898+1
119888119898
119888119898+1minus119899
119888119898+119899
119888119898+119899minus1
119888119898
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(7)
119902 (119909) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 119909 119909119899
119888119898+1
119888119898
119888119898+1minus119899
119888119898+119899
119888119898+119899minus1
119888119898
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(8)
Let us now multiply 119895th row in 119901(119909) and 119902(119909) by 119909119895+119898minus1 (119895 =2 119899 + 1) and afterwards divide 119895th column in 119901(119909) and119902(119909) by 119909119895minus1 (119895 = 2 119899+1) This results in a multiplicationof numerator and denominator by 119909119898119899 Having done so weget
119901 (119909)
119902 (119909)=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
sum119898
119894=0119888119894119909119894
sum119898minus1
119894=0119888119894119909119894
sdotsdotsdot sum119898minus119899
119894=0119888119894119909119894
119888119898+1119909119898+1
119888119898119909119898
sdotsdotsdot 119888119898+1minus119899119909119898+1minus119899
119888119898+119899119909119898+119899
119888119898+119899minus1119909119898+119899minus1
sdotsdotsdot 119888119898119909119898
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 sdotsdotsdot 1
119888119898+1119909119898+1
119888119898119909119898
sdotsdotsdot 119888119898+1minus119899119909119898+1minus119899
119888119898+119899119909119898+119899
119888119898+119899minus1119909119898+119899minus1
sdotsdotsdot 119888119898119909119898
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(9)
if (119863 = det 119863119898119899
= 0)This quotent of determinants can also immediately be
written down for a bivariate function 119891(119909 119910) The sumsum119896
119894=0119888119894119909119894 shall be replaced by 119896th partial sum of the Taylor
series development of 119891(119909 119910) and the expression 119888119896119909119896 by an
expression that contains all the terms of degree 119896 in 119891(119909 119910)Here a bivariate term 119888
119894119895119909119894
119910119895 is said to be of degree 119894 + 119895 If we
define
119901 (119909 119910)
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119894+119895=0
119888119894119895119909119894
119910119895
119898minus1
sum
119894+119895=0
119888119894119895119909119894
119910119895
sdot sdot sdot
119898minus119899
sum
119894+119895=0
119888119894119895119909119894
119910119895
sum
119894+119895=119898+1
119888119894119895119909119894
119910119895
sum
119894+119895=119898
119888119894119895119909119894
119910119895
sdot sdot sdot sum
119894+119895=119898+1minus119899
119888119894119895119909119894
119910119895
sum
119894+119895=119898+119899
119888119894119895119909119894
119910119895
sum
119894+119895=119898+119899minus1
119888119894119895119909119894
119910119895
sdot sdot sdot sum
119894+119895=119898
119888119894119895119909119894
119910119895
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
Abstract and Applied Analysis 3
119902 (119909 119910)
=
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 sdot sdot sdot 1
sum
119894+119895=119898+1
119888119894119895119909119894
119910119895
sum
119894+119895=119898
119888119894119895119909119894
119910119895
sdot sdot sdot sum
119894+119895=119898+1minus119899
119888119894119895119909119894
119910119895
sum
119894+119895=119898+119899
119888119894119895119909119894
119910119895
sum
119894+119895=119898+119899minus1
119888119894119895119909119894
119910119895
sdot sdot sdot sum
119894+119895=119898
119888119894119895119909119894
119910119895
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(10)
Then it is easy to see that 119901(119909 119910) and 119902(119909 119910) are of the form
119901 (119909 119910) =
119898119899+119898
sum
119894+119895=119898119899
119886119894119895119909119894
119910119895
119902 (119909 119910) =
119898119899+119899
sum
119894+119895=119898119899
119887119894119895119909119894
119910119895
(11)
We know that 119901(119909 119910) and 119902(119909 119910) are called Pade equations[25] So themultivariate Pade approximant of order (119898 119899) for119891(119909 119910) is defined as
119903119898119899(119909 119910) =
119901 (119909 119910)
119902 (119909 119910) (12)
5 Numerical Experiments
In this section two methods ADM and MPA shall beillustrated by two examples All the results are calculatedby using the software Maple12 The full ADM solutions ofexamples can be seen from Odibat and Momani [24]
Example 1 Consider the nonlinear time-fractional advectionpartial differential equation [24]
119863120572
lowast119905119906 (119909 119905) + 119906 (119909 119905) 119906
119909(119909 119905) = 119909 + 119909119905
2
119905 gt 0 119909 isin 119877 0 lt 120572 le 1
(13)
subject to the initial condition
119906 (119909 0) = 0 (14)
Odibat and Momani [24] solved the problem using thedecomposition method and they obtained the followingrecurrence relation [24]
1199060(119909 119905) = 119906 (119909 0) + 119869
120572
(119909 + 1199091199052
)
= 119909(119905120572
Γ (120572 + 1)+
2119905120572+2
Γ (120572 + 3))
119906119895+1(119909 119905) = minus119869
120572
(119860119895) 119895 ge 0
(15)
where 119860119895are the Adomian polynomials for the nonlinear
function 119873 = 119906119906119909 In view of (15) the first few components
of the decomposition series are derived in [24] as follows
1199060(119909 119905) = 119909(
119905120572
Γ (120572 + 1)+
2119905120572+2
Γ (120572 + 3))
1199061(119909 119905) = minus119909(
Γ (2120572 + 1) 1199053120572
Γ(120572 + 1)2
Γ (3120572 + 1)
+4Γ (2120572 + 3) 119905
3120572+2
Γ (120572 + 1) Γ (120572 + 3) Γ (3120572 + 3)
+4Γ (2120572 + 5) 119905
3120572+4
Γ(120572 + 3)2
Γ (3120572 + 5)
)
1199062(119909 119905) = 2119909(
Γ (2120572 + 1) Γ (4120572 + 1) 1199055120572
Γ(120572 + 1)3
Γ (3120572 + 1) Γ (5120572 + 1)
+8Γ (2120572 + 5) Γ (4120572 + 7) 119905
5120572+6
Γ(120572 + 1)3
Γ (3120572 + 5) Γ (5120572 + 7)
+ sdot sdot sdot)
(16)
and so on in this manner the rest of components of thedecomposition series can be obtained [24]
The first three terms of the decomposition series are givenby [24]
119906 (119909 119905)
= 119909(119905120572
Γ (120572 + 1)+
2119905120572+2
Γ (120572 + 3)minus
Γ (2120572 + 1) 1199053120572
Γ(120572 + 1)2
Γ (3120572 + 1)
minus4Γ (2120572 + 3) 119905
3120572+2
Γ (120572 + 1) Γ (120572 + 3) Γ (3120572 + 3)+ sdot sdot sdot)
(17)
For 120572 = 1 (16) is
119906 (119909 119905) = 119909119905 + 01 times 10minus9
1199053
minus 013333333331199091199055
(18)
Now let us calculate the approximate solution of (18) for119898 =4 and 119899 = 2 by using Multivariate Pade approximation Toobtain multivariate Pade equations of (18) for119898 = 4 and 119899 =2 we use (10) By using (10) we obtain
119901 (119909 119905)
=
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119909119905 + 01 times 10minus9
1199053
119909119905 119909119905
0 01 times 10minus9
1199053
0
minus013333333331199091199055
0 01 times 10minus9
1199053
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 01333333333 times 10minus10
times (1199052
+ 07500000002 times 10minus9
) 1199093
1199057
4 Abstract and Applied Analysis
119902 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
0 01 times 10minus9
1199053
0
minus013333333331199091199055
0 01 times 10minus9
1199053
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 01333333333 times 10minus10
times (1199052
+ 07500000002 times 10minus9
) 1199092
1199056
(19)
So the multivariate Pade approximation of order (4 2) for(17) that is
[4 2](119909119905)
=
(1199052
+ 07500000002 times 10minus9
) 119909119905
1199052 + 07500000002 times 10minus9 (20)
For 120572 = 05 (17) is
119906 (119909 119905) = 112837916711990911990505
minus 0957797985011990911990515
+ 0601802222611990911990525
minus 0700560811611990911990535
(21)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1128379167119909119886 minus 095779798501199091198863
+ 060180222261199091198865
minus 070056081161199091198867
(22)
Using (10) to calculate the multivariate Pade equations for(22) we get
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1128379167119909119886 minus 095779798501199091198863
+ 060180222261199091198865
1128379167119909119886 minus 095779798501199091198863
1128379167119909119886 minus 095779798501199091198863
0 060180222261199091198865
0
minus070056081161199091198867
0 060180222261199091198865
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus04907854507 (12014642941198864
minus 12318323471198862
minus 08326662354) 1199093
11988611
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
0 060180222261199091198865
0
minus070056081161199091198867
0 060180222261199091198865
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 04907854507 (07379312378 + 17180584831198862
) 1199093
11988610
(23)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (6 2) for (21) that is
[6 2](119909119905)
= minus (12014642941199052
minus 1231832347119905
minus08326662354) 119909radic119905
times (07379312378 + 1718058483119905)minus1
(24)
For 120572 = 075 (17) is
119906 (119909 119905) = 00000712534544111990911990575
+ 01764791440 times 10minus5
11990911990595
minus 01238343301 times 10minus17
119909119905225
minus 02897967272 times 10minus19
119909119905245
(25)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 00000712534544111990911988615
+ 01764791440 times 10minus5
11990911988619
minus 01238343301 times 10minus17
11990911988645
minus 02897967272 times 10minus19
11990911988649
(26)
Using (10) to calculate the multivariate Pade equations andthen recalling that 11990512 = 119886 we get multivariate Padeapproximation of order (49 2) for (25) that is
[49 2](119909119905)
= minus 08398214310 times 10minus39
1199093
1199051132
times ( minus 000007125345441 minus 01764791440
times10minus5
1199052
+ 01238343301 times 10minus17
11990515
)
times (08398214310 times 10minus39
1199092
11990549
)minus1
(27)
Table 1 Figures 1(a) 1(b) 1(c) 2(a) 2(b) 2(c) and 2(d) showsthe approximate solutions for (13) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 1is for the exact solution 119906(119909 119905) = 119909119905 [24]
Example 2 Consider the nonlinear time-fractional hyper-bolic equation [24]
119863120572
lowast119905119906 (119909 119905) =
120597
120597119909(119906 (119909 119905)
120597119906 (119909 119905)
120597119909)
119905 gt 0 119909 isin 119877 1 lt 120572 le 2
(28)
subject to the initial condition
119906 (119909 0) = 1199092
119906119905(119909 0) = minus2119909
2
(29)
Abstract and Applied Analysis 5
1
05
0
minus05
minus1
minus1
minus05
005
1105
0
minus05
minus1
119905 119909
(a)
08
04
0
minus04
minus08
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(b)
1
05
0
minus05
minus1
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(c)
Figure 1 (a) Exact solution of Example 1 for 120572 = 1 (b) ADM solution of Example 1 for 120572 = 1 (c) Multivariate Pade approximation of ADMsolution for 120572 = 1 in Example 1
Odibat and Momani [24] solved the problem using thedecomposition method and they obtained the followingrecurrence relation in [24]
1199060(119909 119905) = 119906 (119909 0) + 119905119906
119909(119909 0) = 119909
2
(1 minus 2119905)
119906119895+1(119909 119905) = 119869
120572
(119860119895)119909
119895 ge 0
(30)
where 119860119895are the Adomian polynomials for the nonlinear
function119873 = 119906119906119909 In view of (30) the first few components
of the decomposition series are derived in [24] as follows
1199060(119909 119905) = 119909
2
(1 minus 2119905)
1199061(119909 119905) = 6119909
2
(119905120572
Γ (120572 + 1)minus
4119905120572+1
Γ (120572 + 2)+
8119905120572+2
Γ (120572 + 3))
1199062(119909 119905) = 72119909
2
(1199052120572
Γ (2120572 + 1)minus
41199052120572+1
Γ (2120572 + 2)
+81199052120572+2
Γ (2120572 + 3)minus
2Γ (120572 + 2) 1199052120572+1
Γ (120572 + 1) Γ (2120572 + 2))
+ 721199092
(8Γ (120572 + 3) 119905
2120572+2
Γ (120572 + 2) Γ (2120572 + 3)
minus16Γ (120572 + 4) 119905
2120572+3
Γ (120572 + 3) Γ (2120572 + 4))
(31)
6 Abstract and Applied Analysis
030250201501005
0
002004
006008
01 108
0604
020
119905 119909
(a)
030250201501005
002004
006008
01 01008
006004
002
119905119909
(b)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(c)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
1105
0minus05
minus1
119905119909
(d)
Figure 2 (a) ADM solution of Example 1 for 120572 = 05 (b) Multivariate Pade approximation of ADM solution for 120572 = 05 in Example 1 (c)ADM solution of Example 1 for 120572 = 075 (d) Multivariate Pade approximation of ADM solution for 120572 = 075 in Example 1
and so on in this manner the rest of components of thedecomposition series can be obtained
The first three terms of the decomposition series (7) aregiven [24] by
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (119905120572
Γ (120572 + 1)minus
4119905120572+1
Γ (120572 + 2)+
8119905120572+2
Γ (120572 + 3))
+ 721199092
(1199052120572
Γ (2120572 + 1)+ sdot sdot sdot)
(32)
For 120572 = 2 (43) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(050000000001199052
minus 066666666681199053
+033333333341199054
)
+ 30000000001199092
1199054
(33)
Now let us calculate the approximate solution of (33) for119898 =4 and 119899 = 2 by using multivariate Pade approximation To
Abstract and Applied Analysis 7
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(a)
2
25
3
15
1
05
0
02
04
06
08
1 108
0604
02
119905 119909
(b)
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(c)
Figure 3 (a) Exact solution of Example 2 for 120572 = 20 (b) ADM solution of Example 2 for 120572 = 20 (c) Multivariate Pade approximation ofADM solution for 120572 = 20 in Example 2
obtain multivariate Pade equations of (33) for119898 = 4 and 119899 =2 we use (10) By using (10) we obtain
119901 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
(1 minus 2119905) + 30000000001199092
1199052
1199092
(1 minus 2119905) 1199092
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus20000000001199054
(028 times 10minus9
1199052
minus 034 times 10minus9
119905 (minus004999999986) 1199096
119902 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 20000000001199054
(00499999999 + 01000000001119905 + 005000000041199052
) 1199094
(34)
So the multivariate Pade approximation of order (4 2) for(33) that is
[4 2](119909119905)
= minus 1000000000 (028 times 10minus9
1199052
minus 034 times 10minus9
119905
minus004999999986) 1199092
times (00499999999 + 01000000001119905
+005000000041199052
)minus1
(35)
8 Abstract and Applied Analysis
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(a)
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(b)
0008
001
0006
0004
0002
0
119905119909
minus01
minus005
0
0050101
0050
minus005
minus01
(c)
0008
001
0006
0004
0002
0
119905 119909
minus01
minus005
0005
0101005
0
minus005
minus01
(d)
Figure 4 (a) ADM solution of Example 2 for 120572 = 15 (b) Multivariate Pade approximation of ADM solution for 120572 = 15 in Example 2 (c)ADM solution of Example 2 for 120572 = 175 (d) Multivariate Pade approximation of ADM solution for 120572 = 175 in Example 2
For 120572 = 15 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (0752252778211990515
minus 120360444511990525
+0687773968311990535
) + 12000000001199092
11990530
(36)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198862
) + 61199092
times (075225277821198863
minus12036044451198865
+ 068777396831199057
)
+ 12000000001199092
1198866
= 1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
+ 41266438101199092
1198867
+ 12000000001199092
1198866
(37)
Using (10) to calculate multivariate Pade equations of (37) for119898 = 7 and 119899 = 2 we use (10) By using (10) we obtain
Abstract and Applied Analysis 9
Table1Num
ericalvalues
when120572=05120572=075and120572=10for(13)
119909119905
120572=05
120572=075
120572=10
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000111
8860
666
000111
8859998
07125363089times10minus21
07125363089times10minus21
000
009999999987
000
0100
0000
000
000
01002
002
0003138022015
0003138007574
02579676158times10minus18
02579676148times10minus18
000
03999999915
000
0400
0000
000
000
04003
003
000571664
0278
0005716554826
08097412617times10minus17
08097412644times10minus17
000
08999999028
000
0900
0000
000
000
09004
004
0008727882362
0008727584792
09339702880times10minus16
09339702881times10minus16
0001599999454
0001600
0000
0000016
005
005
001209607907
001209530419
06224118481times10minus15
06224118464times10minus15
0002499997917
0002500
0000
0000025
006
006
00157687340
8001576705530
02931772549times10minus14
02931772552times10minus14
0003599993779
0003600
0000
0000036
007
007
001970633078
001970312778
01086905954times10minus13
01086905954times10minus13
000
4899984313
000
4900
0000
00000
49008
008
002387754051
002387197301
03381735603times10minus13
03381735608times10minus13
000
6399965047
000
6400
0000
00000
64009
009
002825661342
002824760190
09203528751times10minus13
09203528751times10minus13
0008099929141
0008100
0000
0000081
01
01
003282181204
003280802591
02253790147times10minus12
02253790145times10minus12
000
99998666
6700100
0000
0000
001
10 Abstract and Applied Analysis
Table2Num
ericalvalues
when120572=15120572=175and120572=20for(28)
119909119905
120572=15
120572=175
120572=20
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000
009844537131
000
009844533110
000
0098116
32420
000
0098116
25316
000
00980296
0500
000
00980296
0486
000
00980296
0494
002
002
000
03889833219
000
03889809228
000
03855443171
000
03855410232
000
038446
75200
000
038446
75123
000
038446
75125
003
003
000
0866
4034309
000
08663775704
000
08529749012
000
08529437792
000
08483364
450
000
08483363176
000
08483363182
004
004
0001527388854
0001527250241
0001492265504
0001492112
228
0001479290880
0001479289940
0001479289941
005
005
0002370102454
0002369595331
0002296248891
0002295720729
0002267578126
0002267573695
0002267573696
006
006
0003393997434
0003392539471
0003258613123
0003257161294
0003204
002880
0003203987182
0003203987184
007
007
000
4599726730
000
4596176215
000
4373819216
000
437040
4791
000
4279895445
000
4279849765
000
4279849769
008
008
0005989109024
000598144
9991
0005637043845
00056298804
000005487083520
0005486968446
0005486968450
009
009
0007565131844
000755006
8937
0007044
140677
0007030367394
000
6817867605
000
6817607941
000
6817607945
01
01
000
9331981000
000
9304
436794
0008591616573
0008566
895894
000826500
0000
0008264
462804
0008264
462810
Abstract and Applied Analysis 11
119901 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
12000000001199092
1199056
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (82357601511198865
+ 08791855311198864
+ 12534276361198863
+ 14034265421198862
+ 1750000001119886 + 1053153890) 1199096
11988610
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
12000000001199092
1198866
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (1053153890 + 1750000001119886 + 35097343221198862
) 1199094
11988610
(38)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (7 2) for (36) that is
[7 2](119909119905)
= (823576015111990552
+ 08791855311199052
+ 125342763611990532
+ 1403426542119905
+1750000001radic119905 + 1053153890) 1199092
times (1053153890 + 1750000001radic119905
+3509734322119905)minus1
(39)
For 120572 = 175 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(06217515726119905175
minus 09043659240119905275
+04823284927119905375
)
+ 61899657151199092
11990535
(40)
For simplicity let 11990514 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198864
)
+ 61199092
(062175157261198867
minus 0904365924011988611
+0482328492711988615
) + 61899657151199092
11988614
= 1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
+ 28939709561199092
11988615
+ 61899657151199092
11988614
(41)
Using (10) to calculate multivariate Pade equations of (41) for119898 = 15 and 119899 = 2 we use (10) By using (10) we obtain
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus38315675561199096
11988628
(minus1 + 21198864
minus 37305094361198867
+ 542619554411988611
)
119902 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 38315675561199094
11988628
(42)
recalling that 11990514 = 119886 we get multivariate Pade approxima-tion of order (15 2) for (40) that is
[15 2](119909119905)
= minus 38315675561199096
1199057
times (minus1 + 2119905 minus 373050943611990574
+5426195544119905114
)
times (38315675561199094
1199057
)minus1
(43)
Table 2 Figures 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) and 4(d)show the approximate solutions for (28) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 2is for the exact solution 119906(119909 119905) = (119909119905 + 1)2 [24]
6 Concluding Remarks
The fundamental goal of this paper has been to constructan approximate solution of nonlinear partial differential
12 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal 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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International Journal 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
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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 Abstract and Applied Analysis
The decomposition method requires that a nonlinear frac-tional differential equation (1) is expressed in terms of opera-tor form as
119863120572
lowast119905119906 (119909 119905) + 119871119906 (119909 119905) + 119873119906 (119909 119905) = 119892 (119909 119905) 119909 gt 0 (2)
where 119871 is a linear operator which might include other frac-tional derivatives of order less than 120572119873 is a nonlinear oper-ator which also might include other fractional derivatives oforder less than 120572 119863120572
lowast119905= 120597120572
120597119905120572 is the Caputo fractional
derivative of order 120572 and 119892(119909 119905) is the source function [24]Applying the operator 119869120572 [3 4 22 23] the inverse of the
operator 119863120572lowast119905 to both sides of (5) Odibat and Momani [24]
obtained
119906 (119909 119905) =
119898minus1
sum
119896=0
120597119896
119906
120597119905119896(119909 0+
)119905119896
119896+ 119869120572
119892 (119909 119905)
minus 119869120572
[119871119906 (119909 119905) + 119873119906 (119909 119905)]
infin
sum
119899=0
119906119899(119909 119905) =
119898minus1
sum
119896=0
120597119896
119906
120597119905119896(119909 0+
)119905119896
119896+ 119869120572
119892 (119909 119905)
minus 119869120572
[119871(
infin
sum
119899=0
119906119899(119909 119905)) +
infin
sum
119899=0
119860119899]
(3)
From this the iterates are determined in [24] by the followingrecursive way
1199060(119909 119905) =
119898minus1
sum
119896=0
120597119896
119906
120597119905119896(119909 0+
)119905119896
119896+ 119869120572
119892 (119909 119905)
1199061(119909 119905) = minus119869
120572
(1198711199060+ 1198600)
1199062(119909 119905) = minus119869
120572
(1198711199061+ 1198601)
119906119899+1(119909 119905) = minus119869
120572
(119871119906119899+ 119860119899)
(4)
4 Multivariate Padeacute Aproximation [25]
Consider the bivariate function 119891(119909 119910) with Taylor seriesdevelopment
119891 (119909 119910) =
infin
sum
119894119895=0
119888119894119895119909119894
119910119895 (5)
around the origin We know that a solution of unvariate Padeapproximation problem for
119891 (119909) =
infin
sum
119894=0
119888119894119909119894 (6)
is given by
119901 (119909) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119894=0
119888119894119909119894
119909
119898minus1
sum
119894=0
119888119894119909119894
119909119899
119898minus119899
sum
119894=0
119888119894119909119894
119888119898+1
119888119898
119888119898+1minus119899
119888119898+119899
119888119898+119899minus1
119888119898
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(7)
119902 (119909) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 119909 119909119899
119888119898+1
119888119898
119888119898+1minus119899
119888119898+119899
119888119898+119899minus1
119888119898
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(8)
Let us now multiply 119895th row in 119901(119909) and 119902(119909) by 119909119895+119898minus1 (119895 =2 119899 + 1) and afterwards divide 119895th column in 119901(119909) and119902(119909) by 119909119895minus1 (119895 = 2 119899+1) This results in a multiplicationof numerator and denominator by 119909119898119899 Having done so weget
119901 (119909)
119902 (119909)=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
sum119898
119894=0119888119894119909119894
sum119898minus1
119894=0119888119894119909119894
sdotsdotsdot sum119898minus119899
119894=0119888119894119909119894
119888119898+1119909119898+1
119888119898119909119898
sdotsdotsdot 119888119898+1minus119899119909119898+1minus119899
119888119898+119899119909119898+119899
119888119898+119899minus1119909119898+119899minus1
sdotsdotsdot 119888119898119909119898
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 sdotsdotsdot 1
119888119898+1119909119898+1
119888119898119909119898
sdotsdotsdot 119888119898+1minus119899119909119898+1minus119899
119888119898+119899119909119898+119899
119888119898+119899minus1119909119898+119899minus1
sdotsdotsdot 119888119898119909119898
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(9)
if (119863 = det 119863119898119899
= 0)This quotent of determinants can also immediately be
written down for a bivariate function 119891(119909 119910) The sumsum119896
119894=0119888119894119909119894 shall be replaced by 119896th partial sum of the Taylor
series development of 119891(119909 119910) and the expression 119888119896119909119896 by an
expression that contains all the terms of degree 119896 in 119891(119909 119910)Here a bivariate term 119888
119894119895119909119894
119910119895 is said to be of degree 119894 + 119895 If we
define
119901 (119909 119910)
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119898
sum
119894+119895=0
119888119894119895119909119894
119910119895
119898minus1
sum
119894+119895=0
119888119894119895119909119894
119910119895
sdot sdot sdot
119898minus119899
sum
119894+119895=0
119888119894119895119909119894
119910119895
sum
119894+119895=119898+1
119888119894119895119909119894
119910119895
sum
119894+119895=119898
119888119894119895119909119894
119910119895
sdot sdot sdot sum
119894+119895=119898+1minus119899
119888119894119895119909119894
119910119895
sum
119894+119895=119898+119899
119888119894119895119909119894
119910119895
sum
119894+119895=119898+119899minus1
119888119894119895119909119894
119910119895
sdot sdot sdot sum
119894+119895=119898
119888119894119895119909119894
119910119895
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
Abstract and Applied Analysis 3
119902 (119909 119910)
=
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 sdot sdot sdot 1
sum
119894+119895=119898+1
119888119894119895119909119894
119910119895
sum
119894+119895=119898
119888119894119895119909119894
119910119895
sdot sdot sdot sum
119894+119895=119898+1minus119899
119888119894119895119909119894
119910119895
sum
119894+119895=119898+119899
119888119894119895119909119894
119910119895
sum
119894+119895=119898+119899minus1
119888119894119895119909119894
119910119895
sdot sdot sdot sum
119894+119895=119898
119888119894119895119909119894
119910119895
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(10)
Then it is easy to see that 119901(119909 119910) and 119902(119909 119910) are of the form
119901 (119909 119910) =
119898119899+119898
sum
119894+119895=119898119899
119886119894119895119909119894
119910119895
119902 (119909 119910) =
119898119899+119899
sum
119894+119895=119898119899
119887119894119895119909119894
119910119895
(11)
We know that 119901(119909 119910) and 119902(119909 119910) are called Pade equations[25] So themultivariate Pade approximant of order (119898 119899) for119891(119909 119910) is defined as
119903119898119899(119909 119910) =
119901 (119909 119910)
119902 (119909 119910) (12)
5 Numerical Experiments
In this section two methods ADM and MPA shall beillustrated by two examples All the results are calculatedby using the software Maple12 The full ADM solutions ofexamples can be seen from Odibat and Momani [24]
Example 1 Consider the nonlinear time-fractional advectionpartial differential equation [24]
119863120572
lowast119905119906 (119909 119905) + 119906 (119909 119905) 119906
119909(119909 119905) = 119909 + 119909119905
2
119905 gt 0 119909 isin 119877 0 lt 120572 le 1
(13)
subject to the initial condition
119906 (119909 0) = 0 (14)
Odibat and Momani [24] solved the problem using thedecomposition method and they obtained the followingrecurrence relation [24]
1199060(119909 119905) = 119906 (119909 0) + 119869
120572
(119909 + 1199091199052
)
= 119909(119905120572
Γ (120572 + 1)+
2119905120572+2
Γ (120572 + 3))
119906119895+1(119909 119905) = minus119869
120572
(119860119895) 119895 ge 0
(15)
where 119860119895are the Adomian polynomials for the nonlinear
function 119873 = 119906119906119909 In view of (15) the first few components
of the decomposition series are derived in [24] as follows
1199060(119909 119905) = 119909(
119905120572
Γ (120572 + 1)+
2119905120572+2
Γ (120572 + 3))
1199061(119909 119905) = minus119909(
Γ (2120572 + 1) 1199053120572
Γ(120572 + 1)2
Γ (3120572 + 1)
+4Γ (2120572 + 3) 119905
3120572+2
Γ (120572 + 1) Γ (120572 + 3) Γ (3120572 + 3)
+4Γ (2120572 + 5) 119905
3120572+4
Γ(120572 + 3)2
Γ (3120572 + 5)
)
1199062(119909 119905) = 2119909(
Γ (2120572 + 1) Γ (4120572 + 1) 1199055120572
Γ(120572 + 1)3
Γ (3120572 + 1) Γ (5120572 + 1)
+8Γ (2120572 + 5) Γ (4120572 + 7) 119905
5120572+6
Γ(120572 + 1)3
Γ (3120572 + 5) Γ (5120572 + 7)
+ sdot sdot sdot)
(16)
and so on in this manner the rest of components of thedecomposition series can be obtained [24]
The first three terms of the decomposition series are givenby [24]
119906 (119909 119905)
= 119909(119905120572
Γ (120572 + 1)+
2119905120572+2
Γ (120572 + 3)minus
Γ (2120572 + 1) 1199053120572
Γ(120572 + 1)2
Γ (3120572 + 1)
minus4Γ (2120572 + 3) 119905
3120572+2
Γ (120572 + 1) Γ (120572 + 3) Γ (3120572 + 3)+ sdot sdot sdot)
(17)
For 120572 = 1 (16) is
119906 (119909 119905) = 119909119905 + 01 times 10minus9
1199053
minus 013333333331199091199055
(18)
Now let us calculate the approximate solution of (18) for119898 =4 and 119899 = 2 by using Multivariate Pade approximation Toobtain multivariate Pade equations of (18) for119898 = 4 and 119899 =2 we use (10) By using (10) we obtain
119901 (119909 119905)
=
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119909119905 + 01 times 10minus9
1199053
119909119905 119909119905
0 01 times 10minus9
1199053
0
minus013333333331199091199055
0 01 times 10minus9
1199053
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 01333333333 times 10minus10
times (1199052
+ 07500000002 times 10minus9
) 1199093
1199057
4 Abstract and Applied Analysis
119902 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
0 01 times 10minus9
1199053
0
minus013333333331199091199055
0 01 times 10minus9
1199053
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 01333333333 times 10minus10
times (1199052
+ 07500000002 times 10minus9
) 1199092
1199056
(19)
So the multivariate Pade approximation of order (4 2) for(17) that is
[4 2](119909119905)
=
(1199052
+ 07500000002 times 10minus9
) 119909119905
1199052 + 07500000002 times 10minus9 (20)
For 120572 = 05 (17) is
119906 (119909 119905) = 112837916711990911990505
minus 0957797985011990911990515
+ 0601802222611990911990525
minus 0700560811611990911990535
(21)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1128379167119909119886 minus 095779798501199091198863
+ 060180222261199091198865
minus 070056081161199091198867
(22)
Using (10) to calculate the multivariate Pade equations for(22) we get
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1128379167119909119886 minus 095779798501199091198863
+ 060180222261199091198865
1128379167119909119886 minus 095779798501199091198863
1128379167119909119886 minus 095779798501199091198863
0 060180222261199091198865
0
minus070056081161199091198867
0 060180222261199091198865
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus04907854507 (12014642941198864
minus 12318323471198862
minus 08326662354) 1199093
11988611
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
0 060180222261199091198865
0
minus070056081161199091198867
0 060180222261199091198865
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 04907854507 (07379312378 + 17180584831198862
) 1199093
11988610
(23)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (6 2) for (21) that is
[6 2](119909119905)
= minus (12014642941199052
minus 1231832347119905
minus08326662354) 119909radic119905
times (07379312378 + 1718058483119905)minus1
(24)
For 120572 = 075 (17) is
119906 (119909 119905) = 00000712534544111990911990575
+ 01764791440 times 10minus5
11990911990595
minus 01238343301 times 10minus17
119909119905225
minus 02897967272 times 10minus19
119909119905245
(25)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 00000712534544111990911988615
+ 01764791440 times 10minus5
11990911988619
minus 01238343301 times 10minus17
11990911988645
minus 02897967272 times 10minus19
11990911988649
(26)
Using (10) to calculate the multivariate Pade equations andthen recalling that 11990512 = 119886 we get multivariate Padeapproximation of order (49 2) for (25) that is
[49 2](119909119905)
= minus 08398214310 times 10minus39
1199093
1199051132
times ( minus 000007125345441 minus 01764791440
times10minus5
1199052
+ 01238343301 times 10minus17
11990515
)
times (08398214310 times 10minus39
1199092
11990549
)minus1
(27)
Table 1 Figures 1(a) 1(b) 1(c) 2(a) 2(b) 2(c) and 2(d) showsthe approximate solutions for (13) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 1is for the exact solution 119906(119909 119905) = 119909119905 [24]
Example 2 Consider the nonlinear time-fractional hyper-bolic equation [24]
119863120572
lowast119905119906 (119909 119905) =
120597
120597119909(119906 (119909 119905)
120597119906 (119909 119905)
120597119909)
119905 gt 0 119909 isin 119877 1 lt 120572 le 2
(28)
subject to the initial condition
119906 (119909 0) = 1199092
119906119905(119909 0) = minus2119909
2
(29)
Abstract and Applied Analysis 5
1
05
0
minus05
minus1
minus1
minus05
005
1105
0
minus05
minus1
119905 119909
(a)
08
04
0
minus04
minus08
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(b)
1
05
0
minus05
minus1
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(c)
Figure 1 (a) Exact solution of Example 1 for 120572 = 1 (b) ADM solution of Example 1 for 120572 = 1 (c) Multivariate Pade approximation of ADMsolution for 120572 = 1 in Example 1
Odibat and Momani [24] solved the problem using thedecomposition method and they obtained the followingrecurrence relation in [24]
1199060(119909 119905) = 119906 (119909 0) + 119905119906
119909(119909 0) = 119909
2
(1 minus 2119905)
119906119895+1(119909 119905) = 119869
120572
(119860119895)119909
119895 ge 0
(30)
where 119860119895are the Adomian polynomials for the nonlinear
function119873 = 119906119906119909 In view of (30) the first few components
of the decomposition series are derived in [24] as follows
1199060(119909 119905) = 119909
2
(1 minus 2119905)
1199061(119909 119905) = 6119909
2
(119905120572
Γ (120572 + 1)minus
4119905120572+1
Γ (120572 + 2)+
8119905120572+2
Γ (120572 + 3))
1199062(119909 119905) = 72119909
2
(1199052120572
Γ (2120572 + 1)minus
41199052120572+1
Γ (2120572 + 2)
+81199052120572+2
Γ (2120572 + 3)minus
2Γ (120572 + 2) 1199052120572+1
Γ (120572 + 1) Γ (2120572 + 2))
+ 721199092
(8Γ (120572 + 3) 119905
2120572+2
Γ (120572 + 2) Γ (2120572 + 3)
minus16Γ (120572 + 4) 119905
2120572+3
Γ (120572 + 3) Γ (2120572 + 4))
(31)
6 Abstract and Applied Analysis
030250201501005
0
002004
006008
01 108
0604
020
119905 119909
(a)
030250201501005
002004
006008
01 01008
006004
002
119905119909
(b)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(c)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
1105
0minus05
minus1
119905119909
(d)
Figure 2 (a) ADM solution of Example 1 for 120572 = 05 (b) Multivariate Pade approximation of ADM solution for 120572 = 05 in Example 1 (c)ADM solution of Example 1 for 120572 = 075 (d) Multivariate Pade approximation of ADM solution for 120572 = 075 in Example 1
and so on in this manner the rest of components of thedecomposition series can be obtained
The first three terms of the decomposition series (7) aregiven [24] by
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (119905120572
Γ (120572 + 1)minus
4119905120572+1
Γ (120572 + 2)+
8119905120572+2
Γ (120572 + 3))
+ 721199092
(1199052120572
Γ (2120572 + 1)+ sdot sdot sdot)
(32)
For 120572 = 2 (43) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(050000000001199052
minus 066666666681199053
+033333333341199054
)
+ 30000000001199092
1199054
(33)
Now let us calculate the approximate solution of (33) for119898 =4 and 119899 = 2 by using multivariate Pade approximation To
Abstract and Applied Analysis 7
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(a)
2
25
3
15
1
05
0
02
04
06
08
1 108
0604
02
119905 119909
(b)
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(c)
Figure 3 (a) Exact solution of Example 2 for 120572 = 20 (b) ADM solution of Example 2 for 120572 = 20 (c) Multivariate Pade approximation ofADM solution for 120572 = 20 in Example 2
obtain multivariate Pade equations of (33) for119898 = 4 and 119899 =2 we use (10) By using (10) we obtain
119901 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
(1 minus 2119905) + 30000000001199092
1199052
1199092
(1 minus 2119905) 1199092
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus20000000001199054
(028 times 10minus9
1199052
minus 034 times 10minus9
119905 (minus004999999986) 1199096
119902 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 20000000001199054
(00499999999 + 01000000001119905 + 005000000041199052
) 1199094
(34)
So the multivariate Pade approximation of order (4 2) for(33) that is
[4 2](119909119905)
= minus 1000000000 (028 times 10minus9
1199052
minus 034 times 10minus9
119905
minus004999999986) 1199092
times (00499999999 + 01000000001119905
+005000000041199052
)minus1
(35)
8 Abstract and Applied Analysis
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(a)
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(b)
0008
001
0006
0004
0002
0
119905119909
minus01
minus005
0
0050101
0050
minus005
minus01
(c)
0008
001
0006
0004
0002
0
119905 119909
minus01
minus005
0005
0101005
0
minus005
minus01
(d)
Figure 4 (a) ADM solution of Example 2 for 120572 = 15 (b) Multivariate Pade approximation of ADM solution for 120572 = 15 in Example 2 (c)ADM solution of Example 2 for 120572 = 175 (d) Multivariate Pade approximation of ADM solution for 120572 = 175 in Example 2
For 120572 = 15 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (0752252778211990515
minus 120360444511990525
+0687773968311990535
) + 12000000001199092
11990530
(36)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198862
) + 61199092
times (075225277821198863
minus12036044451198865
+ 068777396831199057
)
+ 12000000001199092
1198866
= 1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
+ 41266438101199092
1198867
+ 12000000001199092
1198866
(37)
Using (10) to calculate multivariate Pade equations of (37) for119898 = 7 and 119899 = 2 we use (10) By using (10) we obtain
Abstract and Applied Analysis 9
Table1Num
ericalvalues
when120572=05120572=075and120572=10for(13)
119909119905
120572=05
120572=075
120572=10
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000111
8860
666
000111
8859998
07125363089times10minus21
07125363089times10minus21
000
009999999987
000
0100
0000
000
000
01002
002
0003138022015
0003138007574
02579676158times10minus18
02579676148times10minus18
000
03999999915
000
0400
0000
000
000
04003
003
000571664
0278
0005716554826
08097412617times10minus17
08097412644times10minus17
000
08999999028
000
0900
0000
000
000
09004
004
0008727882362
0008727584792
09339702880times10minus16
09339702881times10minus16
0001599999454
0001600
0000
0000016
005
005
001209607907
001209530419
06224118481times10minus15
06224118464times10minus15
0002499997917
0002500
0000
0000025
006
006
00157687340
8001576705530
02931772549times10minus14
02931772552times10minus14
0003599993779
0003600
0000
0000036
007
007
001970633078
001970312778
01086905954times10minus13
01086905954times10minus13
000
4899984313
000
4900
0000
00000
49008
008
002387754051
002387197301
03381735603times10minus13
03381735608times10minus13
000
6399965047
000
6400
0000
00000
64009
009
002825661342
002824760190
09203528751times10minus13
09203528751times10minus13
0008099929141
0008100
0000
0000081
01
01
003282181204
003280802591
02253790147times10minus12
02253790145times10minus12
000
99998666
6700100
0000
0000
001
10 Abstract and Applied Analysis
Table2Num
ericalvalues
when120572=15120572=175and120572=20for(28)
119909119905
120572=15
120572=175
120572=20
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000
009844537131
000
009844533110
000
0098116
32420
000
0098116
25316
000
00980296
0500
000
00980296
0486
000
00980296
0494
002
002
000
03889833219
000
03889809228
000
03855443171
000
03855410232
000
038446
75200
000
038446
75123
000
038446
75125
003
003
000
0866
4034309
000
08663775704
000
08529749012
000
08529437792
000
08483364
450
000
08483363176
000
08483363182
004
004
0001527388854
0001527250241
0001492265504
0001492112
228
0001479290880
0001479289940
0001479289941
005
005
0002370102454
0002369595331
0002296248891
0002295720729
0002267578126
0002267573695
0002267573696
006
006
0003393997434
0003392539471
0003258613123
0003257161294
0003204
002880
0003203987182
0003203987184
007
007
000
4599726730
000
4596176215
000
4373819216
000
437040
4791
000
4279895445
000
4279849765
000
4279849769
008
008
0005989109024
000598144
9991
0005637043845
00056298804
000005487083520
0005486968446
0005486968450
009
009
0007565131844
000755006
8937
0007044
140677
0007030367394
000
6817867605
000
6817607941
000
6817607945
01
01
000
9331981000
000
9304
436794
0008591616573
0008566
895894
000826500
0000
0008264
462804
0008264
462810
Abstract and Applied Analysis 11
119901 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
12000000001199092
1199056
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (82357601511198865
+ 08791855311198864
+ 12534276361198863
+ 14034265421198862
+ 1750000001119886 + 1053153890) 1199096
11988610
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
12000000001199092
1198866
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (1053153890 + 1750000001119886 + 35097343221198862
) 1199094
11988610
(38)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (7 2) for (36) that is
[7 2](119909119905)
= (823576015111990552
+ 08791855311199052
+ 125342763611990532
+ 1403426542119905
+1750000001radic119905 + 1053153890) 1199092
times (1053153890 + 1750000001radic119905
+3509734322119905)minus1
(39)
For 120572 = 175 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(06217515726119905175
minus 09043659240119905275
+04823284927119905375
)
+ 61899657151199092
11990535
(40)
For simplicity let 11990514 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198864
)
+ 61199092
(062175157261198867
minus 0904365924011988611
+0482328492711988615
) + 61899657151199092
11988614
= 1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
+ 28939709561199092
11988615
+ 61899657151199092
11988614
(41)
Using (10) to calculate multivariate Pade equations of (41) for119898 = 15 and 119899 = 2 we use (10) By using (10) we obtain
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus38315675561199096
11988628
(minus1 + 21198864
minus 37305094361198867
+ 542619554411988611
)
119902 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 38315675561199094
11988628
(42)
recalling that 11990514 = 119886 we get multivariate Pade approxima-tion of order (15 2) for (40) that is
[15 2](119909119905)
= minus 38315675561199096
1199057
times (minus1 + 2119905 minus 373050943611990574
+5426195544119905114
)
times (38315675561199094
1199057
)minus1
(43)
Table 2 Figures 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) and 4(d)show the approximate solutions for (28) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 2is for the exact solution 119906(119909 119905) = (119909119905 + 1)2 [24]
6 Concluding Remarks
The fundamental goal of this paper has been to constructan approximate solution of nonlinear partial differential
12 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
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
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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
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
119902 (119909 119910)
=
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 sdot sdot sdot 1
sum
119894+119895=119898+1
119888119894119895119909119894
119910119895
sum
119894+119895=119898
119888119894119895119909119894
119910119895
sdot sdot sdot sum
119894+119895=119898+1minus119899
119888119894119895119909119894
119910119895
sum
119894+119895=119898+119899
119888119894119895119909119894
119910119895
sum
119894+119895=119898+119899minus1
119888119894119895119909119894
119910119895
sdot sdot sdot sum
119894+119895=119898
119888119894119895119909119894
119910119895
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(10)
Then it is easy to see that 119901(119909 119910) and 119902(119909 119910) are of the form
119901 (119909 119910) =
119898119899+119898
sum
119894+119895=119898119899
119886119894119895119909119894
119910119895
119902 (119909 119910) =
119898119899+119899
sum
119894+119895=119898119899
119887119894119895119909119894
119910119895
(11)
We know that 119901(119909 119910) and 119902(119909 119910) are called Pade equations[25] So themultivariate Pade approximant of order (119898 119899) for119891(119909 119910) is defined as
119903119898119899(119909 119910) =
119901 (119909 119910)
119902 (119909 119910) (12)
5 Numerical Experiments
In this section two methods ADM and MPA shall beillustrated by two examples All the results are calculatedby using the software Maple12 The full ADM solutions ofexamples can be seen from Odibat and Momani [24]
Example 1 Consider the nonlinear time-fractional advectionpartial differential equation [24]
119863120572
lowast119905119906 (119909 119905) + 119906 (119909 119905) 119906
119909(119909 119905) = 119909 + 119909119905
2
119905 gt 0 119909 isin 119877 0 lt 120572 le 1
(13)
subject to the initial condition
119906 (119909 0) = 0 (14)
Odibat and Momani [24] solved the problem using thedecomposition method and they obtained the followingrecurrence relation [24]
1199060(119909 119905) = 119906 (119909 0) + 119869
120572
(119909 + 1199091199052
)
= 119909(119905120572
Γ (120572 + 1)+
2119905120572+2
Γ (120572 + 3))
119906119895+1(119909 119905) = minus119869
120572
(119860119895) 119895 ge 0
(15)
where 119860119895are the Adomian polynomials for the nonlinear
function 119873 = 119906119906119909 In view of (15) the first few components
of the decomposition series are derived in [24] as follows
1199060(119909 119905) = 119909(
119905120572
Γ (120572 + 1)+
2119905120572+2
Γ (120572 + 3))
1199061(119909 119905) = minus119909(
Γ (2120572 + 1) 1199053120572
Γ(120572 + 1)2
Γ (3120572 + 1)
+4Γ (2120572 + 3) 119905
3120572+2
Γ (120572 + 1) Γ (120572 + 3) Γ (3120572 + 3)
+4Γ (2120572 + 5) 119905
3120572+4
Γ(120572 + 3)2
Γ (3120572 + 5)
)
1199062(119909 119905) = 2119909(
Γ (2120572 + 1) Γ (4120572 + 1) 1199055120572
Γ(120572 + 1)3
Γ (3120572 + 1) Γ (5120572 + 1)
+8Γ (2120572 + 5) Γ (4120572 + 7) 119905
5120572+6
Γ(120572 + 1)3
Γ (3120572 + 5) Γ (5120572 + 7)
+ sdot sdot sdot)
(16)
and so on in this manner the rest of components of thedecomposition series can be obtained [24]
The first three terms of the decomposition series are givenby [24]
119906 (119909 119905)
= 119909(119905120572
Γ (120572 + 1)+
2119905120572+2
Γ (120572 + 3)minus
Γ (2120572 + 1) 1199053120572
Γ(120572 + 1)2
Γ (3120572 + 1)
minus4Γ (2120572 + 3) 119905
3120572+2
Γ (120572 + 1) Γ (120572 + 3) Γ (3120572 + 3)+ sdot sdot sdot)
(17)
For 120572 = 1 (16) is
119906 (119909 119905) = 119909119905 + 01 times 10minus9
1199053
minus 013333333331199091199055
(18)
Now let us calculate the approximate solution of (18) for119898 =4 and 119899 = 2 by using Multivariate Pade approximation Toobtain multivariate Pade equations of (18) for119898 = 4 and 119899 =2 we use (10) By using (10) we obtain
119901 (119909 119905)
=
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119909119905 + 01 times 10minus9
1199053
119909119905 119909119905
0 01 times 10minus9
1199053
0
minus013333333331199091199055
0 01 times 10minus9
1199053
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 01333333333 times 10minus10
times (1199052
+ 07500000002 times 10minus9
) 1199093
1199057
4 Abstract and Applied Analysis
119902 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
0 01 times 10minus9
1199053
0
minus013333333331199091199055
0 01 times 10minus9
1199053
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 01333333333 times 10minus10
times (1199052
+ 07500000002 times 10minus9
) 1199092
1199056
(19)
So the multivariate Pade approximation of order (4 2) for(17) that is
[4 2](119909119905)
=
(1199052
+ 07500000002 times 10minus9
) 119909119905
1199052 + 07500000002 times 10minus9 (20)
For 120572 = 05 (17) is
119906 (119909 119905) = 112837916711990911990505
minus 0957797985011990911990515
+ 0601802222611990911990525
minus 0700560811611990911990535
(21)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1128379167119909119886 minus 095779798501199091198863
+ 060180222261199091198865
minus 070056081161199091198867
(22)
Using (10) to calculate the multivariate Pade equations for(22) we get
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1128379167119909119886 minus 095779798501199091198863
+ 060180222261199091198865
1128379167119909119886 minus 095779798501199091198863
1128379167119909119886 minus 095779798501199091198863
0 060180222261199091198865
0
minus070056081161199091198867
0 060180222261199091198865
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus04907854507 (12014642941198864
minus 12318323471198862
minus 08326662354) 1199093
11988611
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
0 060180222261199091198865
0
minus070056081161199091198867
0 060180222261199091198865
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 04907854507 (07379312378 + 17180584831198862
) 1199093
11988610
(23)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (6 2) for (21) that is
[6 2](119909119905)
= minus (12014642941199052
minus 1231832347119905
minus08326662354) 119909radic119905
times (07379312378 + 1718058483119905)minus1
(24)
For 120572 = 075 (17) is
119906 (119909 119905) = 00000712534544111990911990575
+ 01764791440 times 10minus5
11990911990595
minus 01238343301 times 10minus17
119909119905225
minus 02897967272 times 10minus19
119909119905245
(25)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 00000712534544111990911988615
+ 01764791440 times 10minus5
11990911988619
minus 01238343301 times 10minus17
11990911988645
minus 02897967272 times 10minus19
11990911988649
(26)
Using (10) to calculate the multivariate Pade equations andthen recalling that 11990512 = 119886 we get multivariate Padeapproximation of order (49 2) for (25) that is
[49 2](119909119905)
= minus 08398214310 times 10minus39
1199093
1199051132
times ( minus 000007125345441 minus 01764791440
times10minus5
1199052
+ 01238343301 times 10minus17
11990515
)
times (08398214310 times 10minus39
1199092
11990549
)minus1
(27)
Table 1 Figures 1(a) 1(b) 1(c) 2(a) 2(b) 2(c) and 2(d) showsthe approximate solutions for (13) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 1is for the exact solution 119906(119909 119905) = 119909119905 [24]
Example 2 Consider the nonlinear time-fractional hyper-bolic equation [24]
119863120572
lowast119905119906 (119909 119905) =
120597
120597119909(119906 (119909 119905)
120597119906 (119909 119905)
120597119909)
119905 gt 0 119909 isin 119877 1 lt 120572 le 2
(28)
subject to the initial condition
119906 (119909 0) = 1199092
119906119905(119909 0) = minus2119909
2
(29)
Abstract and Applied Analysis 5
1
05
0
minus05
minus1
minus1
minus05
005
1105
0
minus05
minus1
119905 119909
(a)
08
04
0
minus04
minus08
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(b)
1
05
0
minus05
minus1
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(c)
Figure 1 (a) Exact solution of Example 1 for 120572 = 1 (b) ADM solution of Example 1 for 120572 = 1 (c) Multivariate Pade approximation of ADMsolution for 120572 = 1 in Example 1
Odibat and Momani [24] solved the problem using thedecomposition method and they obtained the followingrecurrence relation in [24]
1199060(119909 119905) = 119906 (119909 0) + 119905119906
119909(119909 0) = 119909
2
(1 minus 2119905)
119906119895+1(119909 119905) = 119869
120572
(119860119895)119909
119895 ge 0
(30)
where 119860119895are the Adomian polynomials for the nonlinear
function119873 = 119906119906119909 In view of (30) the first few components
of the decomposition series are derived in [24] as follows
1199060(119909 119905) = 119909
2
(1 minus 2119905)
1199061(119909 119905) = 6119909
2
(119905120572
Γ (120572 + 1)minus
4119905120572+1
Γ (120572 + 2)+
8119905120572+2
Γ (120572 + 3))
1199062(119909 119905) = 72119909
2
(1199052120572
Γ (2120572 + 1)minus
41199052120572+1
Γ (2120572 + 2)
+81199052120572+2
Γ (2120572 + 3)minus
2Γ (120572 + 2) 1199052120572+1
Γ (120572 + 1) Γ (2120572 + 2))
+ 721199092
(8Γ (120572 + 3) 119905
2120572+2
Γ (120572 + 2) Γ (2120572 + 3)
minus16Γ (120572 + 4) 119905
2120572+3
Γ (120572 + 3) Γ (2120572 + 4))
(31)
6 Abstract and Applied Analysis
030250201501005
0
002004
006008
01 108
0604
020
119905 119909
(a)
030250201501005
002004
006008
01 01008
006004
002
119905119909
(b)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(c)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
1105
0minus05
minus1
119905119909
(d)
Figure 2 (a) ADM solution of Example 1 for 120572 = 05 (b) Multivariate Pade approximation of ADM solution for 120572 = 05 in Example 1 (c)ADM solution of Example 1 for 120572 = 075 (d) Multivariate Pade approximation of ADM solution for 120572 = 075 in Example 1
and so on in this manner the rest of components of thedecomposition series can be obtained
The first three terms of the decomposition series (7) aregiven [24] by
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (119905120572
Γ (120572 + 1)minus
4119905120572+1
Γ (120572 + 2)+
8119905120572+2
Γ (120572 + 3))
+ 721199092
(1199052120572
Γ (2120572 + 1)+ sdot sdot sdot)
(32)
For 120572 = 2 (43) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(050000000001199052
minus 066666666681199053
+033333333341199054
)
+ 30000000001199092
1199054
(33)
Now let us calculate the approximate solution of (33) for119898 =4 and 119899 = 2 by using multivariate Pade approximation To
Abstract and Applied Analysis 7
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(a)
2
25
3
15
1
05
0
02
04
06
08
1 108
0604
02
119905 119909
(b)
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(c)
Figure 3 (a) Exact solution of Example 2 for 120572 = 20 (b) ADM solution of Example 2 for 120572 = 20 (c) Multivariate Pade approximation ofADM solution for 120572 = 20 in Example 2
obtain multivariate Pade equations of (33) for119898 = 4 and 119899 =2 we use (10) By using (10) we obtain
119901 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
(1 minus 2119905) + 30000000001199092
1199052
1199092
(1 minus 2119905) 1199092
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus20000000001199054
(028 times 10minus9
1199052
minus 034 times 10minus9
119905 (minus004999999986) 1199096
119902 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 20000000001199054
(00499999999 + 01000000001119905 + 005000000041199052
) 1199094
(34)
So the multivariate Pade approximation of order (4 2) for(33) that is
[4 2](119909119905)
= minus 1000000000 (028 times 10minus9
1199052
minus 034 times 10minus9
119905
minus004999999986) 1199092
times (00499999999 + 01000000001119905
+005000000041199052
)minus1
(35)
8 Abstract and Applied Analysis
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(a)
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(b)
0008
001
0006
0004
0002
0
119905119909
minus01
minus005
0
0050101
0050
minus005
minus01
(c)
0008
001
0006
0004
0002
0
119905 119909
minus01
minus005
0005
0101005
0
minus005
minus01
(d)
Figure 4 (a) ADM solution of Example 2 for 120572 = 15 (b) Multivariate Pade approximation of ADM solution for 120572 = 15 in Example 2 (c)ADM solution of Example 2 for 120572 = 175 (d) Multivariate Pade approximation of ADM solution for 120572 = 175 in Example 2
For 120572 = 15 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (0752252778211990515
minus 120360444511990525
+0687773968311990535
) + 12000000001199092
11990530
(36)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198862
) + 61199092
times (075225277821198863
minus12036044451198865
+ 068777396831199057
)
+ 12000000001199092
1198866
= 1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
+ 41266438101199092
1198867
+ 12000000001199092
1198866
(37)
Using (10) to calculate multivariate Pade equations of (37) for119898 = 7 and 119899 = 2 we use (10) By using (10) we obtain
Abstract and Applied Analysis 9
Table1Num
ericalvalues
when120572=05120572=075and120572=10for(13)
119909119905
120572=05
120572=075
120572=10
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000111
8860
666
000111
8859998
07125363089times10minus21
07125363089times10minus21
000
009999999987
000
0100
0000
000
000
01002
002
0003138022015
0003138007574
02579676158times10minus18
02579676148times10minus18
000
03999999915
000
0400
0000
000
000
04003
003
000571664
0278
0005716554826
08097412617times10minus17
08097412644times10minus17
000
08999999028
000
0900
0000
000
000
09004
004
0008727882362
0008727584792
09339702880times10minus16
09339702881times10minus16
0001599999454
0001600
0000
0000016
005
005
001209607907
001209530419
06224118481times10minus15
06224118464times10minus15
0002499997917
0002500
0000
0000025
006
006
00157687340
8001576705530
02931772549times10minus14
02931772552times10minus14
0003599993779
0003600
0000
0000036
007
007
001970633078
001970312778
01086905954times10minus13
01086905954times10minus13
000
4899984313
000
4900
0000
00000
49008
008
002387754051
002387197301
03381735603times10minus13
03381735608times10minus13
000
6399965047
000
6400
0000
00000
64009
009
002825661342
002824760190
09203528751times10minus13
09203528751times10minus13
0008099929141
0008100
0000
0000081
01
01
003282181204
003280802591
02253790147times10minus12
02253790145times10minus12
000
99998666
6700100
0000
0000
001
10 Abstract and Applied Analysis
Table2Num
ericalvalues
when120572=15120572=175and120572=20for(28)
119909119905
120572=15
120572=175
120572=20
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000
009844537131
000
009844533110
000
0098116
32420
000
0098116
25316
000
00980296
0500
000
00980296
0486
000
00980296
0494
002
002
000
03889833219
000
03889809228
000
03855443171
000
03855410232
000
038446
75200
000
038446
75123
000
038446
75125
003
003
000
0866
4034309
000
08663775704
000
08529749012
000
08529437792
000
08483364
450
000
08483363176
000
08483363182
004
004
0001527388854
0001527250241
0001492265504
0001492112
228
0001479290880
0001479289940
0001479289941
005
005
0002370102454
0002369595331
0002296248891
0002295720729
0002267578126
0002267573695
0002267573696
006
006
0003393997434
0003392539471
0003258613123
0003257161294
0003204
002880
0003203987182
0003203987184
007
007
000
4599726730
000
4596176215
000
4373819216
000
437040
4791
000
4279895445
000
4279849765
000
4279849769
008
008
0005989109024
000598144
9991
0005637043845
00056298804
000005487083520
0005486968446
0005486968450
009
009
0007565131844
000755006
8937
0007044
140677
0007030367394
000
6817867605
000
6817607941
000
6817607945
01
01
000
9331981000
000
9304
436794
0008591616573
0008566
895894
000826500
0000
0008264
462804
0008264
462810
Abstract and Applied Analysis 11
119901 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
12000000001199092
1199056
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (82357601511198865
+ 08791855311198864
+ 12534276361198863
+ 14034265421198862
+ 1750000001119886 + 1053153890) 1199096
11988610
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
12000000001199092
1198866
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (1053153890 + 1750000001119886 + 35097343221198862
) 1199094
11988610
(38)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (7 2) for (36) that is
[7 2](119909119905)
= (823576015111990552
+ 08791855311199052
+ 125342763611990532
+ 1403426542119905
+1750000001radic119905 + 1053153890) 1199092
times (1053153890 + 1750000001radic119905
+3509734322119905)minus1
(39)
For 120572 = 175 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(06217515726119905175
minus 09043659240119905275
+04823284927119905375
)
+ 61899657151199092
11990535
(40)
For simplicity let 11990514 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198864
)
+ 61199092
(062175157261198867
minus 0904365924011988611
+0482328492711988615
) + 61899657151199092
11988614
= 1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
+ 28939709561199092
11988615
+ 61899657151199092
11988614
(41)
Using (10) to calculate multivariate Pade equations of (41) for119898 = 15 and 119899 = 2 we use (10) By using (10) we obtain
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus38315675561199096
11988628
(minus1 + 21198864
minus 37305094361198867
+ 542619554411988611
)
119902 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 38315675561199094
11988628
(42)
recalling that 11990514 = 119886 we get multivariate Pade approxima-tion of order (15 2) for (40) that is
[15 2](119909119905)
= minus 38315675561199096
1199057
times (minus1 + 2119905 minus 373050943611990574
+5426195544119905114
)
times (38315675561199094
1199057
)minus1
(43)
Table 2 Figures 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) and 4(d)show the approximate solutions for (28) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 2is for the exact solution 119906(119909 119905) = (119909119905 + 1)2 [24]
6 Concluding Remarks
The fundamental goal of this paper has been to constructan approximate solution of nonlinear partial differential
12 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
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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
4 Abstract and Applied Analysis
119902 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
0 01 times 10minus9
1199053
0
minus013333333331199091199055
0 01 times 10minus9
1199053
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 01333333333 times 10minus10
times (1199052
+ 07500000002 times 10minus9
) 1199092
1199056
(19)
So the multivariate Pade approximation of order (4 2) for(17) that is
[4 2](119909119905)
=
(1199052
+ 07500000002 times 10minus9
) 119909119905
1199052 + 07500000002 times 10minus9 (20)
For 120572 = 05 (17) is
119906 (119909 119905) = 112837916711990911990505
minus 0957797985011990911990515
+ 0601802222611990911990525
minus 0700560811611990911990535
(21)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1128379167119909119886 minus 095779798501199091198863
+ 060180222261199091198865
minus 070056081161199091198867
(22)
Using (10) to calculate the multivariate Pade equations for(22) we get
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1128379167119909119886 minus 095779798501199091198863
+ 060180222261199091198865
1128379167119909119886 minus 095779798501199091198863
1128379167119909119886 minus 095779798501199091198863
0 060180222261199091198865
0
minus070056081161199091198867
0 060180222261199091198865
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus04907854507 (12014642941198864
minus 12318323471198862
minus 08326662354) 1199093
11988611
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
0 060180222261199091198865
0
minus070056081161199091198867
0 060180222261199091198865
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 04907854507 (07379312378 + 17180584831198862
) 1199093
11988610
(23)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (6 2) for (21) that is
[6 2](119909119905)
= minus (12014642941199052
minus 1231832347119905
minus08326662354) 119909radic119905
times (07379312378 + 1718058483119905)minus1
(24)
For 120572 = 075 (17) is
119906 (119909 119905) = 00000712534544111990911990575
+ 01764791440 times 10minus5
11990911990595
minus 01238343301 times 10minus17
119909119905225
minus 02897967272 times 10minus19
119909119905245
(25)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 00000712534544111990911988615
+ 01764791440 times 10minus5
11990911988619
minus 01238343301 times 10minus17
11990911988645
minus 02897967272 times 10minus19
11990911988649
(26)
Using (10) to calculate the multivariate Pade equations andthen recalling that 11990512 = 119886 we get multivariate Padeapproximation of order (49 2) for (25) that is
[49 2](119909119905)
= minus 08398214310 times 10minus39
1199093
1199051132
times ( minus 000007125345441 minus 01764791440
times10minus5
1199052
+ 01238343301 times 10minus17
11990515
)
times (08398214310 times 10minus39
1199092
11990549
)minus1
(27)
Table 1 Figures 1(a) 1(b) 1(c) 2(a) 2(b) 2(c) and 2(d) showsthe approximate solutions for (13) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 1is for the exact solution 119906(119909 119905) = 119909119905 [24]
Example 2 Consider the nonlinear time-fractional hyper-bolic equation [24]
119863120572
lowast119905119906 (119909 119905) =
120597
120597119909(119906 (119909 119905)
120597119906 (119909 119905)
120597119909)
119905 gt 0 119909 isin 119877 1 lt 120572 le 2
(28)
subject to the initial condition
119906 (119909 0) = 1199092
119906119905(119909 0) = minus2119909
2
(29)
Abstract and Applied Analysis 5
1
05
0
minus05
minus1
minus1
minus05
005
1105
0
minus05
minus1
119905 119909
(a)
08
04
0
minus04
minus08
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(b)
1
05
0
minus05
minus1
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(c)
Figure 1 (a) Exact solution of Example 1 for 120572 = 1 (b) ADM solution of Example 1 for 120572 = 1 (c) Multivariate Pade approximation of ADMsolution for 120572 = 1 in Example 1
Odibat and Momani [24] solved the problem using thedecomposition method and they obtained the followingrecurrence relation in [24]
1199060(119909 119905) = 119906 (119909 0) + 119905119906
119909(119909 0) = 119909
2
(1 minus 2119905)
119906119895+1(119909 119905) = 119869
120572
(119860119895)119909
119895 ge 0
(30)
where 119860119895are the Adomian polynomials for the nonlinear
function119873 = 119906119906119909 In view of (30) the first few components
of the decomposition series are derived in [24] as follows
1199060(119909 119905) = 119909
2
(1 minus 2119905)
1199061(119909 119905) = 6119909
2
(119905120572
Γ (120572 + 1)minus
4119905120572+1
Γ (120572 + 2)+
8119905120572+2
Γ (120572 + 3))
1199062(119909 119905) = 72119909
2
(1199052120572
Γ (2120572 + 1)minus
41199052120572+1
Γ (2120572 + 2)
+81199052120572+2
Γ (2120572 + 3)minus
2Γ (120572 + 2) 1199052120572+1
Γ (120572 + 1) Γ (2120572 + 2))
+ 721199092
(8Γ (120572 + 3) 119905
2120572+2
Γ (120572 + 2) Γ (2120572 + 3)
minus16Γ (120572 + 4) 119905
2120572+3
Γ (120572 + 3) Γ (2120572 + 4))
(31)
6 Abstract and Applied Analysis
030250201501005
0
002004
006008
01 108
0604
020
119905 119909
(a)
030250201501005
002004
006008
01 01008
006004
002
119905119909
(b)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(c)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
1105
0minus05
minus1
119905119909
(d)
Figure 2 (a) ADM solution of Example 1 for 120572 = 05 (b) Multivariate Pade approximation of ADM solution for 120572 = 05 in Example 1 (c)ADM solution of Example 1 for 120572 = 075 (d) Multivariate Pade approximation of ADM solution for 120572 = 075 in Example 1
and so on in this manner the rest of components of thedecomposition series can be obtained
The first three terms of the decomposition series (7) aregiven [24] by
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (119905120572
Γ (120572 + 1)minus
4119905120572+1
Γ (120572 + 2)+
8119905120572+2
Γ (120572 + 3))
+ 721199092
(1199052120572
Γ (2120572 + 1)+ sdot sdot sdot)
(32)
For 120572 = 2 (43) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(050000000001199052
minus 066666666681199053
+033333333341199054
)
+ 30000000001199092
1199054
(33)
Now let us calculate the approximate solution of (33) for119898 =4 and 119899 = 2 by using multivariate Pade approximation To
Abstract and Applied Analysis 7
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(a)
2
25
3
15
1
05
0
02
04
06
08
1 108
0604
02
119905 119909
(b)
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(c)
Figure 3 (a) Exact solution of Example 2 for 120572 = 20 (b) ADM solution of Example 2 for 120572 = 20 (c) Multivariate Pade approximation ofADM solution for 120572 = 20 in Example 2
obtain multivariate Pade equations of (33) for119898 = 4 and 119899 =2 we use (10) By using (10) we obtain
119901 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
(1 minus 2119905) + 30000000001199092
1199052
1199092
(1 minus 2119905) 1199092
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus20000000001199054
(028 times 10minus9
1199052
minus 034 times 10minus9
119905 (minus004999999986) 1199096
119902 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 20000000001199054
(00499999999 + 01000000001119905 + 005000000041199052
) 1199094
(34)
So the multivariate Pade approximation of order (4 2) for(33) that is
[4 2](119909119905)
= minus 1000000000 (028 times 10minus9
1199052
minus 034 times 10minus9
119905
minus004999999986) 1199092
times (00499999999 + 01000000001119905
+005000000041199052
)minus1
(35)
8 Abstract and Applied Analysis
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(a)
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(b)
0008
001
0006
0004
0002
0
119905119909
minus01
minus005
0
0050101
0050
minus005
minus01
(c)
0008
001
0006
0004
0002
0
119905 119909
minus01
minus005
0005
0101005
0
minus005
minus01
(d)
Figure 4 (a) ADM solution of Example 2 for 120572 = 15 (b) Multivariate Pade approximation of ADM solution for 120572 = 15 in Example 2 (c)ADM solution of Example 2 for 120572 = 175 (d) Multivariate Pade approximation of ADM solution for 120572 = 175 in Example 2
For 120572 = 15 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (0752252778211990515
minus 120360444511990525
+0687773968311990535
) + 12000000001199092
11990530
(36)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198862
) + 61199092
times (075225277821198863
minus12036044451198865
+ 068777396831199057
)
+ 12000000001199092
1198866
= 1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
+ 41266438101199092
1198867
+ 12000000001199092
1198866
(37)
Using (10) to calculate multivariate Pade equations of (37) for119898 = 7 and 119899 = 2 we use (10) By using (10) we obtain
Abstract and Applied Analysis 9
Table1Num
ericalvalues
when120572=05120572=075and120572=10for(13)
119909119905
120572=05
120572=075
120572=10
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000111
8860
666
000111
8859998
07125363089times10minus21
07125363089times10minus21
000
009999999987
000
0100
0000
000
000
01002
002
0003138022015
0003138007574
02579676158times10minus18
02579676148times10minus18
000
03999999915
000
0400
0000
000
000
04003
003
000571664
0278
0005716554826
08097412617times10minus17
08097412644times10minus17
000
08999999028
000
0900
0000
000
000
09004
004
0008727882362
0008727584792
09339702880times10minus16
09339702881times10minus16
0001599999454
0001600
0000
0000016
005
005
001209607907
001209530419
06224118481times10minus15
06224118464times10minus15
0002499997917
0002500
0000
0000025
006
006
00157687340
8001576705530
02931772549times10minus14
02931772552times10minus14
0003599993779
0003600
0000
0000036
007
007
001970633078
001970312778
01086905954times10minus13
01086905954times10minus13
000
4899984313
000
4900
0000
00000
49008
008
002387754051
002387197301
03381735603times10minus13
03381735608times10minus13
000
6399965047
000
6400
0000
00000
64009
009
002825661342
002824760190
09203528751times10minus13
09203528751times10minus13
0008099929141
0008100
0000
0000081
01
01
003282181204
003280802591
02253790147times10minus12
02253790145times10minus12
000
99998666
6700100
0000
0000
001
10 Abstract and Applied Analysis
Table2Num
ericalvalues
when120572=15120572=175and120572=20for(28)
119909119905
120572=15
120572=175
120572=20
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000
009844537131
000
009844533110
000
0098116
32420
000
0098116
25316
000
00980296
0500
000
00980296
0486
000
00980296
0494
002
002
000
03889833219
000
03889809228
000
03855443171
000
03855410232
000
038446
75200
000
038446
75123
000
038446
75125
003
003
000
0866
4034309
000
08663775704
000
08529749012
000
08529437792
000
08483364
450
000
08483363176
000
08483363182
004
004
0001527388854
0001527250241
0001492265504
0001492112
228
0001479290880
0001479289940
0001479289941
005
005
0002370102454
0002369595331
0002296248891
0002295720729
0002267578126
0002267573695
0002267573696
006
006
0003393997434
0003392539471
0003258613123
0003257161294
0003204
002880
0003203987182
0003203987184
007
007
000
4599726730
000
4596176215
000
4373819216
000
437040
4791
000
4279895445
000
4279849765
000
4279849769
008
008
0005989109024
000598144
9991
0005637043845
00056298804
000005487083520
0005486968446
0005486968450
009
009
0007565131844
000755006
8937
0007044
140677
0007030367394
000
6817867605
000
6817607941
000
6817607945
01
01
000
9331981000
000
9304
436794
0008591616573
0008566
895894
000826500
0000
0008264
462804
0008264
462810
Abstract and Applied Analysis 11
119901 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
12000000001199092
1199056
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (82357601511198865
+ 08791855311198864
+ 12534276361198863
+ 14034265421198862
+ 1750000001119886 + 1053153890) 1199096
11988610
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
12000000001199092
1198866
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (1053153890 + 1750000001119886 + 35097343221198862
) 1199094
11988610
(38)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (7 2) for (36) that is
[7 2](119909119905)
= (823576015111990552
+ 08791855311199052
+ 125342763611990532
+ 1403426542119905
+1750000001radic119905 + 1053153890) 1199092
times (1053153890 + 1750000001radic119905
+3509734322119905)minus1
(39)
For 120572 = 175 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(06217515726119905175
minus 09043659240119905275
+04823284927119905375
)
+ 61899657151199092
11990535
(40)
For simplicity let 11990514 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198864
)
+ 61199092
(062175157261198867
minus 0904365924011988611
+0482328492711988615
) + 61899657151199092
11988614
= 1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
+ 28939709561199092
11988615
+ 61899657151199092
11988614
(41)
Using (10) to calculate multivariate Pade equations of (41) for119898 = 15 and 119899 = 2 we use (10) By using (10) we obtain
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus38315675561199096
11988628
(minus1 + 21198864
minus 37305094361198867
+ 542619554411988611
)
119902 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 38315675561199094
11988628
(42)
recalling that 11990514 = 119886 we get multivariate Pade approxima-tion of order (15 2) for (40) that is
[15 2](119909119905)
= minus 38315675561199096
1199057
times (minus1 + 2119905 minus 373050943611990574
+5426195544119905114
)
times (38315675561199094
1199057
)minus1
(43)
Table 2 Figures 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) and 4(d)show the approximate solutions for (28) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 2is for the exact solution 119906(119909 119905) = (119909119905 + 1)2 [24]
6 Concluding Remarks
The fundamental goal of this paper has been to constructan approximate solution of nonlinear partial differential
12 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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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
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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
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom 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
Abstract and Applied Analysis 5
1
05
0
minus05
minus1
minus1
minus05
005
1105
0
minus05
minus1
119905 119909
(a)
08
04
0
minus04
minus08
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(b)
1
05
0
minus05
minus1
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(c)
Figure 1 (a) Exact solution of Example 1 for 120572 = 1 (b) ADM solution of Example 1 for 120572 = 1 (c) Multivariate Pade approximation of ADMsolution for 120572 = 1 in Example 1
Odibat and Momani [24] solved the problem using thedecomposition method and they obtained the followingrecurrence relation in [24]
1199060(119909 119905) = 119906 (119909 0) + 119905119906
119909(119909 0) = 119909
2
(1 minus 2119905)
119906119895+1(119909 119905) = 119869
120572
(119860119895)119909
119895 ge 0
(30)
where 119860119895are the Adomian polynomials for the nonlinear
function119873 = 119906119906119909 In view of (30) the first few components
of the decomposition series are derived in [24] as follows
1199060(119909 119905) = 119909
2
(1 minus 2119905)
1199061(119909 119905) = 6119909
2
(119905120572
Γ (120572 + 1)minus
4119905120572+1
Γ (120572 + 2)+
8119905120572+2
Γ (120572 + 3))
1199062(119909 119905) = 72119909
2
(1199052120572
Γ (2120572 + 1)minus
41199052120572+1
Γ (2120572 + 2)
+81199052120572+2
Γ (2120572 + 3)minus
2Γ (120572 + 2) 1199052120572+1
Γ (120572 + 1) Γ (2120572 + 2))
+ 721199092
(8Γ (120572 + 3) 119905
2120572+2
Γ (120572 + 2) Γ (2120572 + 3)
minus16Γ (120572 + 4) 119905
2120572+3
Γ (120572 + 3) Γ (2120572 + 4))
(31)
6 Abstract and Applied Analysis
030250201501005
0
002004
006008
01 108
0604
020
119905 119909
(a)
030250201501005
002004
006008
01 01008
006004
002
119905119909
(b)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(c)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
1105
0minus05
minus1
119905119909
(d)
Figure 2 (a) ADM solution of Example 1 for 120572 = 05 (b) Multivariate Pade approximation of ADM solution for 120572 = 05 in Example 1 (c)ADM solution of Example 1 for 120572 = 075 (d) Multivariate Pade approximation of ADM solution for 120572 = 075 in Example 1
and so on in this manner the rest of components of thedecomposition series can be obtained
The first three terms of the decomposition series (7) aregiven [24] by
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (119905120572
Γ (120572 + 1)minus
4119905120572+1
Γ (120572 + 2)+
8119905120572+2
Γ (120572 + 3))
+ 721199092
(1199052120572
Γ (2120572 + 1)+ sdot sdot sdot)
(32)
For 120572 = 2 (43) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(050000000001199052
minus 066666666681199053
+033333333341199054
)
+ 30000000001199092
1199054
(33)
Now let us calculate the approximate solution of (33) for119898 =4 and 119899 = 2 by using multivariate Pade approximation To
Abstract and Applied Analysis 7
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(a)
2
25
3
15
1
05
0
02
04
06
08
1 108
0604
02
119905 119909
(b)
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(c)
Figure 3 (a) Exact solution of Example 2 for 120572 = 20 (b) ADM solution of Example 2 for 120572 = 20 (c) Multivariate Pade approximation ofADM solution for 120572 = 20 in Example 2
obtain multivariate Pade equations of (33) for119898 = 4 and 119899 =2 we use (10) By using (10) we obtain
119901 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
(1 minus 2119905) + 30000000001199092
1199052
1199092
(1 minus 2119905) 1199092
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus20000000001199054
(028 times 10minus9
1199052
minus 034 times 10minus9
119905 (minus004999999986) 1199096
119902 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 20000000001199054
(00499999999 + 01000000001119905 + 005000000041199052
) 1199094
(34)
So the multivariate Pade approximation of order (4 2) for(33) that is
[4 2](119909119905)
= minus 1000000000 (028 times 10minus9
1199052
minus 034 times 10minus9
119905
minus004999999986) 1199092
times (00499999999 + 01000000001119905
+005000000041199052
)minus1
(35)
8 Abstract and Applied Analysis
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(a)
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(b)
0008
001
0006
0004
0002
0
119905119909
minus01
minus005
0
0050101
0050
minus005
minus01
(c)
0008
001
0006
0004
0002
0
119905 119909
minus01
minus005
0005
0101005
0
minus005
minus01
(d)
Figure 4 (a) ADM solution of Example 2 for 120572 = 15 (b) Multivariate Pade approximation of ADM solution for 120572 = 15 in Example 2 (c)ADM solution of Example 2 for 120572 = 175 (d) Multivariate Pade approximation of ADM solution for 120572 = 175 in Example 2
For 120572 = 15 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (0752252778211990515
minus 120360444511990525
+0687773968311990535
) + 12000000001199092
11990530
(36)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198862
) + 61199092
times (075225277821198863
minus12036044451198865
+ 068777396831199057
)
+ 12000000001199092
1198866
= 1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
+ 41266438101199092
1198867
+ 12000000001199092
1198866
(37)
Using (10) to calculate multivariate Pade equations of (37) for119898 = 7 and 119899 = 2 we use (10) By using (10) we obtain
Abstract and Applied Analysis 9
Table1Num
ericalvalues
when120572=05120572=075and120572=10for(13)
119909119905
120572=05
120572=075
120572=10
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000111
8860
666
000111
8859998
07125363089times10minus21
07125363089times10minus21
000
009999999987
000
0100
0000
000
000
01002
002
0003138022015
0003138007574
02579676158times10minus18
02579676148times10minus18
000
03999999915
000
0400
0000
000
000
04003
003
000571664
0278
0005716554826
08097412617times10minus17
08097412644times10minus17
000
08999999028
000
0900
0000
000
000
09004
004
0008727882362
0008727584792
09339702880times10minus16
09339702881times10minus16
0001599999454
0001600
0000
0000016
005
005
001209607907
001209530419
06224118481times10minus15
06224118464times10minus15
0002499997917
0002500
0000
0000025
006
006
00157687340
8001576705530
02931772549times10minus14
02931772552times10minus14
0003599993779
0003600
0000
0000036
007
007
001970633078
001970312778
01086905954times10minus13
01086905954times10minus13
000
4899984313
000
4900
0000
00000
49008
008
002387754051
002387197301
03381735603times10minus13
03381735608times10minus13
000
6399965047
000
6400
0000
00000
64009
009
002825661342
002824760190
09203528751times10minus13
09203528751times10minus13
0008099929141
0008100
0000
0000081
01
01
003282181204
003280802591
02253790147times10minus12
02253790145times10minus12
000
99998666
6700100
0000
0000
001
10 Abstract and Applied Analysis
Table2Num
ericalvalues
when120572=15120572=175and120572=20for(28)
119909119905
120572=15
120572=175
120572=20
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000
009844537131
000
009844533110
000
0098116
32420
000
0098116
25316
000
00980296
0500
000
00980296
0486
000
00980296
0494
002
002
000
03889833219
000
03889809228
000
03855443171
000
03855410232
000
038446
75200
000
038446
75123
000
038446
75125
003
003
000
0866
4034309
000
08663775704
000
08529749012
000
08529437792
000
08483364
450
000
08483363176
000
08483363182
004
004
0001527388854
0001527250241
0001492265504
0001492112
228
0001479290880
0001479289940
0001479289941
005
005
0002370102454
0002369595331
0002296248891
0002295720729
0002267578126
0002267573695
0002267573696
006
006
0003393997434
0003392539471
0003258613123
0003257161294
0003204
002880
0003203987182
0003203987184
007
007
000
4599726730
000
4596176215
000
4373819216
000
437040
4791
000
4279895445
000
4279849765
000
4279849769
008
008
0005989109024
000598144
9991
0005637043845
00056298804
000005487083520
0005486968446
0005486968450
009
009
0007565131844
000755006
8937
0007044
140677
0007030367394
000
6817867605
000
6817607941
000
6817607945
01
01
000
9331981000
000
9304
436794
0008591616573
0008566
895894
000826500
0000
0008264
462804
0008264
462810
Abstract and Applied Analysis 11
119901 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
12000000001199092
1199056
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (82357601511198865
+ 08791855311198864
+ 12534276361198863
+ 14034265421198862
+ 1750000001119886 + 1053153890) 1199096
11988610
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
12000000001199092
1198866
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (1053153890 + 1750000001119886 + 35097343221198862
) 1199094
11988610
(38)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (7 2) for (36) that is
[7 2](119909119905)
= (823576015111990552
+ 08791855311199052
+ 125342763611990532
+ 1403426542119905
+1750000001radic119905 + 1053153890) 1199092
times (1053153890 + 1750000001radic119905
+3509734322119905)minus1
(39)
For 120572 = 175 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(06217515726119905175
minus 09043659240119905275
+04823284927119905375
)
+ 61899657151199092
11990535
(40)
For simplicity let 11990514 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198864
)
+ 61199092
(062175157261198867
minus 0904365924011988611
+0482328492711988615
) + 61899657151199092
11988614
= 1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
+ 28939709561199092
11988615
+ 61899657151199092
11988614
(41)
Using (10) to calculate multivariate Pade equations of (41) for119898 = 15 and 119899 = 2 we use (10) By using (10) we obtain
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus38315675561199096
11988628
(minus1 + 21198864
minus 37305094361198867
+ 542619554411988611
)
119902 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 38315675561199094
11988628
(42)
recalling that 11990514 = 119886 we get multivariate Pade approxima-tion of order (15 2) for (40) that is
[15 2](119909119905)
= minus 38315675561199096
1199057
times (minus1 + 2119905 minus 373050943611990574
+5426195544119905114
)
times (38315675561199094
1199057
)minus1
(43)
Table 2 Figures 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) and 4(d)show the approximate solutions for (28) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 2is for the exact solution 119906(119909 119905) = (119909119905 + 1)2 [24]
6 Concluding Remarks
The fundamental goal of this paper has been to constructan approximate solution of nonlinear partial differential
12 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
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
6 Abstract and Applied Analysis
030250201501005
0
002004
006008
01 108
0604
020
119905 119909
(a)
030250201501005
002004
006008
01 01008
006004
002
119905119909
(b)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
11
050
minus05
minus1
119905 119909
(c)
6119890minus05
4119890minus05
2119890minus05
0minus2119890minus05
minus4119890minus05
minus6119890minus05
minus1
minus05
005
1105
0minus05
minus1
119905119909
(d)
Figure 2 (a) ADM solution of Example 1 for 120572 = 05 (b) Multivariate Pade approximation of ADM solution for 120572 = 05 in Example 1 (c)ADM solution of Example 1 for 120572 = 075 (d) Multivariate Pade approximation of ADM solution for 120572 = 075 in Example 1
and so on in this manner the rest of components of thedecomposition series can be obtained
The first three terms of the decomposition series (7) aregiven [24] by
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (119905120572
Γ (120572 + 1)minus
4119905120572+1
Γ (120572 + 2)+
8119905120572+2
Γ (120572 + 3))
+ 721199092
(1199052120572
Γ (2120572 + 1)+ sdot sdot sdot)
(32)
For 120572 = 2 (43) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(050000000001199052
minus 066666666681199053
+033333333341199054
)
+ 30000000001199092
1199054
(33)
Now let us calculate the approximate solution of (33) for119898 =4 and 119899 = 2 by using multivariate Pade approximation To
Abstract and Applied Analysis 7
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(a)
2
25
3
15
1
05
0
02
04
06
08
1 108
0604
02
119905 119909
(b)
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(c)
Figure 3 (a) Exact solution of Example 2 for 120572 = 20 (b) ADM solution of Example 2 for 120572 = 20 (c) Multivariate Pade approximation ofADM solution for 120572 = 20 in Example 2
obtain multivariate Pade equations of (33) for119898 = 4 and 119899 =2 we use (10) By using (10) we obtain
119901 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
(1 minus 2119905) + 30000000001199092
1199052
1199092
(1 minus 2119905) 1199092
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus20000000001199054
(028 times 10minus9
1199052
minus 034 times 10minus9
119905 (minus004999999986) 1199096
119902 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 20000000001199054
(00499999999 + 01000000001119905 + 005000000041199052
) 1199094
(34)
So the multivariate Pade approximation of order (4 2) for(33) that is
[4 2](119909119905)
= minus 1000000000 (028 times 10minus9
1199052
minus 034 times 10minus9
119905
minus004999999986) 1199092
times (00499999999 + 01000000001119905
+005000000041199052
)minus1
(35)
8 Abstract and Applied Analysis
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(a)
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(b)
0008
001
0006
0004
0002
0
119905119909
minus01
minus005
0
0050101
0050
minus005
minus01
(c)
0008
001
0006
0004
0002
0
119905 119909
minus01
minus005
0005
0101005
0
minus005
minus01
(d)
Figure 4 (a) ADM solution of Example 2 for 120572 = 15 (b) Multivariate Pade approximation of ADM solution for 120572 = 15 in Example 2 (c)ADM solution of Example 2 for 120572 = 175 (d) Multivariate Pade approximation of ADM solution for 120572 = 175 in Example 2
For 120572 = 15 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (0752252778211990515
minus 120360444511990525
+0687773968311990535
) + 12000000001199092
11990530
(36)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198862
) + 61199092
times (075225277821198863
minus12036044451198865
+ 068777396831199057
)
+ 12000000001199092
1198866
= 1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
+ 41266438101199092
1198867
+ 12000000001199092
1198866
(37)
Using (10) to calculate multivariate Pade equations of (37) for119898 = 7 and 119899 = 2 we use (10) By using (10) we obtain
Abstract and Applied Analysis 9
Table1Num
ericalvalues
when120572=05120572=075and120572=10for(13)
119909119905
120572=05
120572=075
120572=10
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000111
8860
666
000111
8859998
07125363089times10minus21
07125363089times10minus21
000
009999999987
000
0100
0000
000
000
01002
002
0003138022015
0003138007574
02579676158times10minus18
02579676148times10minus18
000
03999999915
000
0400
0000
000
000
04003
003
000571664
0278
0005716554826
08097412617times10minus17
08097412644times10minus17
000
08999999028
000
0900
0000
000
000
09004
004
0008727882362
0008727584792
09339702880times10minus16
09339702881times10minus16
0001599999454
0001600
0000
0000016
005
005
001209607907
001209530419
06224118481times10minus15
06224118464times10minus15
0002499997917
0002500
0000
0000025
006
006
00157687340
8001576705530
02931772549times10minus14
02931772552times10minus14
0003599993779
0003600
0000
0000036
007
007
001970633078
001970312778
01086905954times10minus13
01086905954times10minus13
000
4899984313
000
4900
0000
00000
49008
008
002387754051
002387197301
03381735603times10minus13
03381735608times10minus13
000
6399965047
000
6400
0000
00000
64009
009
002825661342
002824760190
09203528751times10minus13
09203528751times10minus13
0008099929141
0008100
0000
0000081
01
01
003282181204
003280802591
02253790147times10minus12
02253790145times10minus12
000
99998666
6700100
0000
0000
001
10 Abstract and Applied Analysis
Table2Num
ericalvalues
when120572=15120572=175and120572=20for(28)
119909119905
120572=15
120572=175
120572=20
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000
009844537131
000
009844533110
000
0098116
32420
000
0098116
25316
000
00980296
0500
000
00980296
0486
000
00980296
0494
002
002
000
03889833219
000
03889809228
000
03855443171
000
03855410232
000
038446
75200
000
038446
75123
000
038446
75125
003
003
000
0866
4034309
000
08663775704
000
08529749012
000
08529437792
000
08483364
450
000
08483363176
000
08483363182
004
004
0001527388854
0001527250241
0001492265504
0001492112
228
0001479290880
0001479289940
0001479289941
005
005
0002370102454
0002369595331
0002296248891
0002295720729
0002267578126
0002267573695
0002267573696
006
006
0003393997434
0003392539471
0003258613123
0003257161294
0003204
002880
0003203987182
0003203987184
007
007
000
4599726730
000
4596176215
000
4373819216
000
437040
4791
000
4279895445
000
4279849765
000
4279849769
008
008
0005989109024
000598144
9991
0005637043845
00056298804
000005487083520
0005486968446
0005486968450
009
009
0007565131844
000755006
8937
0007044
140677
0007030367394
000
6817867605
000
6817607941
000
6817607945
01
01
000
9331981000
000
9304
436794
0008591616573
0008566
895894
000826500
0000
0008264
462804
0008264
462810
Abstract and Applied Analysis 11
119901 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
12000000001199092
1199056
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (82357601511198865
+ 08791855311198864
+ 12534276361198863
+ 14034265421198862
+ 1750000001119886 + 1053153890) 1199096
11988610
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
12000000001199092
1198866
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (1053153890 + 1750000001119886 + 35097343221198862
) 1199094
11988610
(38)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (7 2) for (36) that is
[7 2](119909119905)
= (823576015111990552
+ 08791855311199052
+ 125342763611990532
+ 1403426542119905
+1750000001radic119905 + 1053153890) 1199092
times (1053153890 + 1750000001radic119905
+3509734322119905)minus1
(39)
For 120572 = 175 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(06217515726119905175
minus 09043659240119905275
+04823284927119905375
)
+ 61899657151199092
11990535
(40)
For simplicity let 11990514 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198864
)
+ 61199092
(062175157261198867
minus 0904365924011988611
+0482328492711988615
) + 61899657151199092
11988614
= 1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
+ 28939709561199092
11988615
+ 61899657151199092
11988614
(41)
Using (10) to calculate multivariate Pade equations of (41) for119898 = 15 and 119899 = 2 we use (10) By using (10) we obtain
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus38315675561199096
11988628
(minus1 + 21198864
minus 37305094361198867
+ 542619554411988611
)
119902 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 38315675561199094
11988628
(42)
recalling that 11990514 = 119886 we get multivariate Pade approxima-tion of order (15 2) for (40) that is
[15 2](119909119905)
= minus 38315675561199096
1199057
times (minus1 + 2119905 minus 373050943611990574
+5426195544119905114
)
times (38315675561199094
1199057
)minus1
(43)
Table 2 Figures 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) and 4(d)show the approximate solutions for (28) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 2is for the exact solution 119906(119909 119905) = (119909119905 + 1)2 [24]
6 Concluding Remarks
The fundamental goal of this paper has been to constructan approximate solution of nonlinear partial differential
12 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
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
Abstract and Applied Analysis 7
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(a)
2
25
3
15
1
05
0
02
04
06
08
1 108
0604
02
119905 119909
(b)
08
06
04
02
0
02
04
06
08
1 108
0604
02
119905 119909
(c)
Figure 3 (a) Exact solution of Example 2 for 120572 = 20 (b) ADM solution of Example 2 for 120572 = 20 (c) Multivariate Pade approximation ofADM solution for 120572 = 20 in Example 2
obtain multivariate Pade equations of (33) for119898 = 4 and 119899 =2 we use (10) By using (10) we obtain
119901 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
(1 minus 2119905) + 30000000001199092
1199052
1199092
(1 minus 2119905) 1199092
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus20000000001199054
(028 times 10minus9
1199052
minus 034 times 10minus9
119905 (minus004999999986) 1199096
119902 (119909 119905)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
minus40000000011199092
1199053
30000000001199092
1199052
minus21199092
119905
50000000001199092
1199054
minus40000000011199092
1199053
30000000001199092
1199052
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 20000000001199054
(00499999999 + 01000000001119905 + 005000000041199052
) 1199094
(34)
So the multivariate Pade approximation of order (4 2) for(33) that is
[4 2](119909119905)
= minus 1000000000 (028 times 10minus9
1199052
minus 034 times 10minus9
119905
minus004999999986) 1199092
times (00499999999 + 01000000001119905
+005000000041199052
)minus1
(35)
8 Abstract and Applied Analysis
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(a)
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(b)
0008
001
0006
0004
0002
0
119905119909
minus01
minus005
0
0050101
0050
minus005
minus01
(c)
0008
001
0006
0004
0002
0
119905 119909
minus01
minus005
0005
0101005
0
minus005
minus01
(d)
Figure 4 (a) ADM solution of Example 2 for 120572 = 15 (b) Multivariate Pade approximation of ADM solution for 120572 = 15 in Example 2 (c)ADM solution of Example 2 for 120572 = 175 (d) Multivariate Pade approximation of ADM solution for 120572 = 175 in Example 2
For 120572 = 15 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (0752252778211990515
minus 120360444511990525
+0687773968311990535
) + 12000000001199092
11990530
(36)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198862
) + 61199092
times (075225277821198863
minus12036044451198865
+ 068777396831199057
)
+ 12000000001199092
1198866
= 1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
+ 41266438101199092
1198867
+ 12000000001199092
1198866
(37)
Using (10) to calculate multivariate Pade equations of (37) for119898 = 7 and 119899 = 2 we use (10) By using (10) we obtain
Abstract and Applied Analysis 9
Table1Num
ericalvalues
when120572=05120572=075and120572=10for(13)
119909119905
120572=05
120572=075
120572=10
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000111
8860
666
000111
8859998
07125363089times10minus21
07125363089times10minus21
000
009999999987
000
0100
0000
000
000
01002
002
0003138022015
0003138007574
02579676158times10minus18
02579676148times10minus18
000
03999999915
000
0400
0000
000
000
04003
003
000571664
0278
0005716554826
08097412617times10minus17
08097412644times10minus17
000
08999999028
000
0900
0000
000
000
09004
004
0008727882362
0008727584792
09339702880times10minus16
09339702881times10minus16
0001599999454
0001600
0000
0000016
005
005
001209607907
001209530419
06224118481times10minus15
06224118464times10minus15
0002499997917
0002500
0000
0000025
006
006
00157687340
8001576705530
02931772549times10minus14
02931772552times10minus14
0003599993779
0003600
0000
0000036
007
007
001970633078
001970312778
01086905954times10minus13
01086905954times10minus13
000
4899984313
000
4900
0000
00000
49008
008
002387754051
002387197301
03381735603times10minus13
03381735608times10minus13
000
6399965047
000
6400
0000
00000
64009
009
002825661342
002824760190
09203528751times10minus13
09203528751times10minus13
0008099929141
0008100
0000
0000081
01
01
003282181204
003280802591
02253790147times10minus12
02253790145times10minus12
000
99998666
6700100
0000
0000
001
10 Abstract and Applied Analysis
Table2Num
ericalvalues
when120572=15120572=175and120572=20for(28)
119909119905
120572=15
120572=175
120572=20
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000
009844537131
000
009844533110
000
0098116
32420
000
0098116
25316
000
00980296
0500
000
00980296
0486
000
00980296
0494
002
002
000
03889833219
000
03889809228
000
03855443171
000
03855410232
000
038446
75200
000
038446
75123
000
038446
75125
003
003
000
0866
4034309
000
08663775704
000
08529749012
000
08529437792
000
08483364
450
000
08483363176
000
08483363182
004
004
0001527388854
0001527250241
0001492265504
0001492112
228
0001479290880
0001479289940
0001479289941
005
005
0002370102454
0002369595331
0002296248891
0002295720729
0002267578126
0002267573695
0002267573696
006
006
0003393997434
0003392539471
0003258613123
0003257161294
0003204
002880
0003203987182
0003203987184
007
007
000
4599726730
000
4596176215
000
4373819216
000
437040
4791
000
4279895445
000
4279849765
000
4279849769
008
008
0005989109024
000598144
9991
0005637043845
00056298804
000005487083520
0005486968446
0005486968450
009
009
0007565131844
000755006
8937
0007044
140677
0007030367394
000
6817867605
000
6817607941
000
6817607945
01
01
000
9331981000
000
9304
436794
0008591616573
0008566
895894
000826500
0000
0008264
462804
0008264
462810
Abstract and Applied Analysis 11
119901 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
12000000001199092
1199056
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (82357601511198865
+ 08791855311198864
+ 12534276361198863
+ 14034265421198862
+ 1750000001119886 + 1053153890) 1199096
11988610
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
12000000001199092
1198866
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (1053153890 + 1750000001119886 + 35097343221198862
) 1199094
11988610
(38)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (7 2) for (36) that is
[7 2](119909119905)
= (823576015111990552
+ 08791855311199052
+ 125342763611990532
+ 1403426542119905
+1750000001radic119905 + 1053153890) 1199092
times (1053153890 + 1750000001radic119905
+3509734322119905)minus1
(39)
For 120572 = 175 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(06217515726119905175
minus 09043659240119905275
+04823284927119905375
)
+ 61899657151199092
11990535
(40)
For simplicity let 11990514 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198864
)
+ 61199092
(062175157261198867
minus 0904365924011988611
+0482328492711988615
) + 61899657151199092
11988614
= 1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
+ 28939709561199092
11988615
+ 61899657151199092
11988614
(41)
Using (10) to calculate multivariate Pade equations of (41) for119898 = 15 and 119899 = 2 we use (10) By using (10) we obtain
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus38315675561199096
11988628
(minus1 + 21198864
minus 37305094361198867
+ 542619554411988611
)
119902 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 38315675561199094
11988628
(42)
recalling that 11990514 = 119886 we get multivariate Pade approxima-tion of order (15 2) for (40) that is
[15 2](119909119905)
= minus 38315675561199096
1199057
times (minus1 + 2119905 minus 373050943611990574
+5426195544119905114
)
times (38315675561199094
1199057
)minus1
(43)
Table 2 Figures 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) and 4(d)show the approximate solutions for (28) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 2is for the exact solution 119906(119909 119905) = (119909119905 + 1)2 [24]
6 Concluding Remarks
The fundamental goal of this paper has been to constructan approximate solution of nonlinear partial differential
12 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
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
8 Abstract and Applied Analysis
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(a)
0008
001
0006
0004
0002
0
002
004
006
008
01 01008
006004
002
119905 119909
(b)
0008
001
0006
0004
0002
0
119905119909
minus01
minus005
0
0050101
0050
minus005
minus01
(c)
0008
001
0006
0004
0002
0
119905 119909
minus01
minus005
0005
0101005
0
minus005
minus01
(d)
Figure 4 (a) ADM solution of Example 2 for 120572 = 15 (b) Multivariate Pade approximation of ADM solution for 120572 = 15 in Example 2 (c)ADM solution of Example 2 for 120572 = 175 (d) Multivariate Pade approximation of ADM solution for 120572 = 175 in Example 2
For 120572 = 15 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905) + 61199092
times (0752252778211990515
minus 120360444511990525
+0687773968311990535
) + 12000000001199092
11990530
(36)
For simplicity let 11990512 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198862
) + 61199092
times (075225277821198863
minus12036044451198865
+ 068777396831199057
)
+ 12000000001199092
1198866
= 1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
+ 41266438101199092
1198867
+ 12000000001199092
1198866
(37)
Using (10) to calculate multivariate Pade equations of (37) for119898 = 7 and 119899 = 2 we use (10) By using (10) we obtain
Abstract and Applied Analysis 9
Table1Num
ericalvalues
when120572=05120572=075and120572=10for(13)
119909119905
120572=05
120572=075
120572=10
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000111
8860
666
000111
8859998
07125363089times10minus21
07125363089times10minus21
000
009999999987
000
0100
0000
000
000
01002
002
0003138022015
0003138007574
02579676158times10minus18
02579676148times10minus18
000
03999999915
000
0400
0000
000
000
04003
003
000571664
0278
0005716554826
08097412617times10minus17
08097412644times10minus17
000
08999999028
000
0900
0000
000
000
09004
004
0008727882362
0008727584792
09339702880times10minus16
09339702881times10minus16
0001599999454
0001600
0000
0000016
005
005
001209607907
001209530419
06224118481times10minus15
06224118464times10minus15
0002499997917
0002500
0000
0000025
006
006
00157687340
8001576705530
02931772549times10minus14
02931772552times10minus14
0003599993779
0003600
0000
0000036
007
007
001970633078
001970312778
01086905954times10minus13
01086905954times10minus13
000
4899984313
000
4900
0000
00000
49008
008
002387754051
002387197301
03381735603times10minus13
03381735608times10minus13
000
6399965047
000
6400
0000
00000
64009
009
002825661342
002824760190
09203528751times10minus13
09203528751times10minus13
0008099929141
0008100
0000
0000081
01
01
003282181204
003280802591
02253790147times10minus12
02253790145times10minus12
000
99998666
6700100
0000
0000
001
10 Abstract and Applied Analysis
Table2Num
ericalvalues
when120572=15120572=175and120572=20for(28)
119909119905
120572=15
120572=175
120572=20
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000
009844537131
000
009844533110
000
0098116
32420
000
0098116
25316
000
00980296
0500
000
00980296
0486
000
00980296
0494
002
002
000
03889833219
000
03889809228
000
03855443171
000
03855410232
000
038446
75200
000
038446
75123
000
038446
75125
003
003
000
0866
4034309
000
08663775704
000
08529749012
000
08529437792
000
08483364
450
000
08483363176
000
08483363182
004
004
0001527388854
0001527250241
0001492265504
0001492112
228
0001479290880
0001479289940
0001479289941
005
005
0002370102454
0002369595331
0002296248891
0002295720729
0002267578126
0002267573695
0002267573696
006
006
0003393997434
0003392539471
0003258613123
0003257161294
0003204
002880
0003203987182
0003203987184
007
007
000
4599726730
000
4596176215
000
4373819216
000
437040
4791
000
4279895445
000
4279849765
000
4279849769
008
008
0005989109024
000598144
9991
0005637043845
00056298804
000005487083520
0005486968446
0005486968450
009
009
0007565131844
000755006
8937
0007044
140677
0007030367394
000
6817867605
000
6817607941
000
6817607945
01
01
000
9331981000
000
9304
436794
0008591616573
0008566
895894
000826500
0000
0008264
462804
0008264
462810
Abstract and Applied Analysis 11
119901 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
12000000001199092
1199056
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (82357601511198865
+ 08791855311198864
+ 12534276361198863
+ 14034265421198862
+ 1750000001119886 + 1053153890) 1199096
11988610
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
12000000001199092
1198866
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (1053153890 + 1750000001119886 + 35097343221198862
) 1199094
11988610
(38)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (7 2) for (36) that is
[7 2](119909119905)
= (823576015111990552
+ 08791855311199052
+ 125342763611990532
+ 1403426542119905
+1750000001radic119905 + 1053153890) 1199092
times (1053153890 + 1750000001radic119905
+3509734322119905)minus1
(39)
For 120572 = 175 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(06217515726119905175
minus 09043659240119905275
+04823284927119905375
)
+ 61899657151199092
11990535
(40)
For simplicity let 11990514 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198864
)
+ 61199092
(062175157261198867
minus 0904365924011988611
+0482328492711988615
) + 61899657151199092
11988614
= 1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
+ 28939709561199092
11988615
+ 61899657151199092
11988614
(41)
Using (10) to calculate multivariate Pade equations of (41) for119898 = 15 and 119899 = 2 we use (10) By using (10) we obtain
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus38315675561199096
11988628
(minus1 + 21198864
minus 37305094361198867
+ 542619554411988611
)
119902 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 38315675561199094
11988628
(42)
recalling that 11990514 = 119886 we get multivariate Pade approxima-tion of order (15 2) for (40) that is
[15 2](119909119905)
= minus 38315675561199096
1199057
times (minus1 + 2119905 minus 373050943611990574
+5426195544119905114
)
times (38315675561199094
1199057
)minus1
(43)
Table 2 Figures 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) and 4(d)show the approximate solutions for (28) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 2is for the exact solution 119906(119909 119905) = (119909119905 + 1)2 [24]
6 Concluding Remarks
The fundamental goal of this paper has been to constructan approximate solution of nonlinear partial differential
12 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
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
Abstract and Applied Analysis 9
Table1Num
ericalvalues
when120572=05120572=075and120572=10for(13)
119909119905
120572=05
120572=075
120572=10
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000111
8860
666
000111
8859998
07125363089times10minus21
07125363089times10minus21
000
009999999987
000
0100
0000
000
000
01002
002
0003138022015
0003138007574
02579676158times10minus18
02579676148times10minus18
000
03999999915
000
0400
0000
000
000
04003
003
000571664
0278
0005716554826
08097412617times10minus17
08097412644times10minus17
000
08999999028
000
0900
0000
000
000
09004
004
0008727882362
0008727584792
09339702880times10minus16
09339702881times10minus16
0001599999454
0001600
0000
0000016
005
005
001209607907
001209530419
06224118481times10minus15
06224118464times10minus15
0002499997917
0002500
0000
0000025
006
006
00157687340
8001576705530
02931772549times10minus14
02931772552times10minus14
0003599993779
0003600
0000
0000036
007
007
001970633078
001970312778
01086905954times10minus13
01086905954times10minus13
000
4899984313
000
4900
0000
00000
49008
008
002387754051
002387197301
03381735603times10minus13
03381735608times10minus13
000
6399965047
000
6400
0000
00000
64009
009
002825661342
002824760190
09203528751times10minus13
09203528751times10minus13
0008099929141
0008100
0000
0000081
01
01
003282181204
003280802591
02253790147times10minus12
02253790145times10minus12
000
99998666
6700100
0000
0000
001
10 Abstract and Applied Analysis
Table2Num
ericalvalues
when120572=15120572=175and120572=20for(28)
119909119905
120572=15
120572=175
120572=20
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000
009844537131
000
009844533110
000
0098116
32420
000
0098116
25316
000
00980296
0500
000
00980296
0486
000
00980296
0494
002
002
000
03889833219
000
03889809228
000
03855443171
000
03855410232
000
038446
75200
000
038446
75123
000
038446
75125
003
003
000
0866
4034309
000
08663775704
000
08529749012
000
08529437792
000
08483364
450
000
08483363176
000
08483363182
004
004
0001527388854
0001527250241
0001492265504
0001492112
228
0001479290880
0001479289940
0001479289941
005
005
0002370102454
0002369595331
0002296248891
0002295720729
0002267578126
0002267573695
0002267573696
006
006
0003393997434
0003392539471
0003258613123
0003257161294
0003204
002880
0003203987182
0003203987184
007
007
000
4599726730
000
4596176215
000
4373819216
000
437040
4791
000
4279895445
000
4279849765
000
4279849769
008
008
0005989109024
000598144
9991
0005637043845
00056298804
000005487083520
0005486968446
0005486968450
009
009
0007565131844
000755006
8937
0007044
140677
0007030367394
000
6817867605
000
6817607941
000
6817607945
01
01
000
9331981000
000
9304
436794
0008591616573
0008566
895894
000826500
0000
0008264
462804
0008264
462810
Abstract and Applied Analysis 11
119901 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
12000000001199092
1199056
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (82357601511198865
+ 08791855311198864
+ 12534276361198863
+ 14034265421198862
+ 1750000001119886 + 1053153890) 1199096
11988610
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
12000000001199092
1198866
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (1053153890 + 1750000001119886 + 35097343221198862
) 1199094
11988610
(38)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (7 2) for (36) that is
[7 2](119909119905)
= (823576015111990552
+ 08791855311199052
+ 125342763611990532
+ 1403426542119905
+1750000001radic119905 + 1053153890) 1199092
times (1053153890 + 1750000001radic119905
+3509734322119905)minus1
(39)
For 120572 = 175 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(06217515726119905175
minus 09043659240119905275
+04823284927119905375
)
+ 61899657151199092
11990535
(40)
For simplicity let 11990514 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198864
)
+ 61199092
(062175157261198867
minus 0904365924011988611
+0482328492711988615
) + 61899657151199092
11988614
= 1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
+ 28939709561199092
11988615
+ 61899657151199092
11988614
(41)
Using (10) to calculate multivariate Pade equations of (41) for119898 = 15 and 119899 = 2 we use (10) By using (10) we obtain
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus38315675561199096
11988628
(minus1 + 21198864
minus 37305094361198867
+ 542619554411988611
)
119902 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 38315675561199094
11988628
(42)
recalling that 11990514 = 119886 we get multivariate Pade approxima-tion of order (15 2) for (40) that is
[15 2](119909119905)
= minus 38315675561199096
1199057
times (minus1 + 2119905 minus 373050943611990574
+5426195544119905114
)
times (38315675561199094
1199057
)minus1
(43)
Table 2 Figures 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) and 4(d)show the approximate solutions for (28) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 2is for the exact solution 119906(119909 119905) = (119909119905 + 1)2 [24]
6 Concluding Remarks
The fundamental goal of this paper has been to constructan approximate solution of nonlinear partial differential
12 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
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
10 Abstract and Applied Analysis
Table2Num
ericalvalues
when120572=15120572=175and120572=20for(28)
119909119905
120572=15
120572=175
120572=20
119906ADM
119906MPA
119906ADM
119906MPA
119906ADM
119906MPA
119906Ex
act
001
001
000
009844537131
000
009844533110
000
0098116
32420
000
0098116
25316
000
00980296
0500
000
00980296
0486
000
00980296
0494
002
002
000
03889833219
000
03889809228
000
03855443171
000
03855410232
000
038446
75200
000
038446
75123
000
038446
75125
003
003
000
0866
4034309
000
08663775704
000
08529749012
000
08529437792
000
08483364
450
000
08483363176
000
08483363182
004
004
0001527388854
0001527250241
0001492265504
0001492112
228
0001479290880
0001479289940
0001479289941
005
005
0002370102454
0002369595331
0002296248891
0002295720729
0002267578126
0002267573695
0002267573696
006
006
0003393997434
0003392539471
0003258613123
0003257161294
0003204
002880
0003203987182
0003203987184
007
007
000
4599726730
000
4596176215
000
4373819216
000
437040
4791
000
4279895445
000
4279849765
000
4279849769
008
008
0005989109024
000598144
9991
0005637043845
00056298804
000005487083520
0005486968446
0005486968450
009
009
0007565131844
000755006
8937
0007044
140677
0007030367394
000
6817867605
000
6817607941
000
6817607945
01
01
000
9331981000
000
9304
436794
0008591616573
0008566
895894
000826500
0000
0008264
462804
0008264
462810
Abstract and Applied Analysis 11
119901 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
12000000001199092
1199056
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (82357601511198865
+ 08791855311198864
+ 12534276361198863
+ 14034265421198862
+ 1750000001119886 + 1053153890) 1199096
11988610
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
12000000001199092
1198866
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (1053153890 + 1750000001119886 + 35097343221198862
) 1199094
11988610
(38)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (7 2) for (36) that is
[7 2](119909119905)
= (823576015111990552
+ 08791855311199052
+ 125342763611990532
+ 1403426542119905
+1750000001radic119905 + 1053153890) 1199092
times (1053153890 + 1750000001radic119905
+3509734322119905)minus1
(39)
For 120572 = 175 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(06217515726119905175
minus 09043659240119905275
+04823284927119905375
)
+ 61899657151199092
11990535
(40)
For simplicity let 11990514 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198864
)
+ 61199092
(062175157261198867
minus 0904365924011988611
+0482328492711988615
) + 61899657151199092
11988614
= 1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
+ 28939709561199092
11988615
+ 61899657151199092
11988614
(41)
Using (10) to calculate multivariate Pade equations of (41) for119898 = 15 and 119899 = 2 we use (10) By using (10) we obtain
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus38315675561199096
11988628
(minus1 + 21198864
minus 37305094361198867
+ 542619554411988611
)
119902 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 38315675561199094
11988628
(42)
recalling that 11990514 = 119886 we get multivariate Pade approxima-tion of order (15 2) for (40) that is
[15 2](119909119905)
= minus 38315675561199096
1199057
times (minus1 + 2119905 minus 373050943611990574
+5426195544119905114
)
times (38315675561199094
1199057
)minus1
(43)
Table 2 Figures 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) and 4(d)show the approximate solutions for (28) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 2is for the exact solution 119906(119909 119905) = (119909119905 + 1)2 [24]
6 Concluding Remarks
The fundamental goal of this paper has been to constructan approximate solution of nonlinear partial differential
12 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
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
Abstract and Applied Analysis 11
119901 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
minus 72216266701199092
1198865
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
1199092
minus 1199092
21198862
+ 45135166691199092
1198863
12000000001199092
1199056
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (82357601511198865
+ 08791855311198864
+ 12534276361198863
+ 14034265421198862
+ 1750000001119886 + 1053153890) 1199096
11988610
119902 (119909 119886) =
10038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
12000000001199092
1198866
minus72216266701199092
1198865
0
41266438101199092
1198867
12000000001199092
1198866
minus72216266701199092
119886
10038161003816100381610038161003816100381610038161003816100381610038161003816
= 4951972572 (1053153890 + 1750000001119886 + 35097343221198862
) 1199094
11988610
(38)
recalling that 11990512 = 119886 we get multivariate Pade approxima-tion of order (7 2) for (36) that is
[7 2](119909119905)
= (823576015111990552
+ 08791855311199052
+ 125342763611990532
+ 1403426542119905
+1750000001radic119905 + 1053153890) 1199092
times (1053153890 + 1750000001radic119905
+3509734322119905)minus1
(39)
For 120572 = 175 (32) is
119906 (119909 119905) = 1199092
(1 minus 2119905)
+ 61199092
(06217515726119905175
minus 09043659240119905275
+04823284927119905375
)
+ 61899657151199092
11990535
(40)
For simplicity let 11990514 = 119886 then
119906 (119909 119886) = 1199092
(1 minus 21198864
)
+ 61199092
(062175157261198867
minus 0904365924011988611
+0482328492711988615
) + 61899657151199092
11988614
= 1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
+ 28939709561199092
11988615
+ 61899657151199092
11988614
(41)
Using (10) to calculate multivariate Pade equations of (41) for119898 = 15 and 119899 = 2 we use (10) By using (10) we obtain
119901 (119909 119886)
=
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
1199092
minus 21199092
1198864
+ 37305094361199092
1198867
minus 54261955441199092
11988611
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= minus38315675561199096
11988628
(minus1 + 21198864
minus 37305094361198867
+ 542619554411988611
)
119902 (119909 119886) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1 1 1
61899657151199092
11988614
0 0
28939709561199092
11988615
61899657151199092
11988614
0
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 38315675561199094
11988628
(42)
recalling that 11990514 = 119886 we get multivariate Pade approxima-tion of order (15 2) for (40) that is
[15 2](119909119905)
= minus 38315675561199096
1199057
times (minus1 + 2119905 minus 373050943611990574
+5426195544119905114
)
times (38315675561199094
1199057
)minus1
(43)
Table 2 Figures 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) and 4(d)show the approximate solutions for (28) obtained for differentvalues of 120572 using the decomposition method (ADM) and themultivariate Pade approximation (MPA) The value of 120572 = 2is for the exact solution 119906(119909 119905) = (119909119905 + 1)2 [24]
6 Concluding Remarks
The fundamental goal of this paper has been to constructan approximate solution of nonlinear partial differential
12 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
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 Abstract and Applied Analysis
equations of fractional order by using multivariate PadeapproximationThe goal has been achieved by using the mul-tivariate Pade approximation and comparing with the Ado-mian decomposition method The present work shows thevalidity and great potential of the multivariate Pade approxi-mation for solving nonlinear partial differential equations offractional order from the numerical resultsNumerical resultsobtained using the multivariate Pade approximation and theAdomian decompositionmethod are in agreement with exactsolutions
References
[1] J H He ldquoNonlinear oscillation with fractional derivative andits applicationsrdquo in Proceedings of International Conference onVibrating Engineering pp 288ndash291 Dalian China 1998
[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science andTechnology vol 15 no 2 pp 86ndash90 1999
[3] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativeSeries A08-98 Fachbreich Mathematik und Informatik FreicUniversitat Berlin Berlin Germany 1998
[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999
[5] SMomani ldquoNon-perturbative analytical solutions of the space-and time-fractional Burgers equationsrdquo Chaos Solitons amp Frac-tals vol 28 no 4 pp 930ndash937 2006
[6] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006
[7] S Momani and Z Odibat ldquoAnalytical solution of a time-fractional Navier-Stokes equation by Adomian decompositionmethodrdquo Applied Mathematics and Computation vol 177 no 2pp 488ndash494 2006
[8] S Momani and Z Odibat ldquoNumerical comparison of methodsfor solving linear differential equations of fractional orderrdquoChaos Solitons amp Fractals vol 31 no 5 pp 1248ndash1255 2007
[9] Z M Odibat and S Momani ldquoApproximate solutions forboundary value problems of time-fractional wave equationrdquoApplied Mathematics and Computation vol 181 no 1 pp 767ndash774 2006
[10] G Domairry and N Nadim ldquoAssessment of homotopy analysismethod and homotopy perturbationmethod in non-linear heattransfer equationrdquo International Communications in Heat andMass Transfer vol 35 no 1 pp 93ndash102 2008
[11] G Domairry M Ahangari and M Jamshidi ldquoExact andanalytical solution for nonlinear dispersive 119870(119898 119901) equationsusing homotopy perturbation methodrdquo Physics Letters A vol368 no 3-4 pp 266ndash270 2007
[12] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[13] G Adomian Solving Frontier Problems of Physics The Decom-position Method vol 60 of Fundamental Theories of PhysicsKluwer Academic Publishers Dordrecht The Netherlands1994
[14] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005
[15] Ph Guillaume and A Huard ldquoMultivariate Pade approxima-tionrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 197ndash219 2000
[16] V Turut E Celik and M Yigider ldquoMultivariate Pade approx-imation for solving partial differential equations (PDE)rdquo Inter-national Journal for Numerical Methods in Fluids vol 66 no 9pp 1159ndash1173 2011
[17] V Turut and N Guzel ldquoComparing numerical methods forsolving time-fractional reaction-diffusion equationsrdquo ISRNMathematical Analysis vol 2012 Article ID 737206 28 pages2012
[18] V Turut ldquoApplication of Multivariate pade approximation forpartial differential equationsrdquo Batman University Journal of LifeSciences vol 2 no 1 pp 17ndash28 2012
[19] V Turut and N Guzel ldquoOn solving partial differential eqau-ations of fractional order by using the variational iterationmethod and multivariate pade approximationrdquo European Jour-nal of Pure and Applied Mathematics Accepted
[20] S Momani and R Qaralleh ldquoNumerical approximations andPade approximants for a fractional population growth modelrdquoApplied Mathematical Modelling vol 31 no 9 pp 1907ndash19142007
[21] S Momani and N Shawagfeh ldquoDecomposition method forsolving fractional Riccati differential equationsrdquo Applied Math-ematics and Computation vol 182 no 2 pp 1083ndash1092 2006
[22] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974
[23] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independent Part IIrdquo Journal of the Royal AstronomicalSociety vol 13 no 5 pp 529ndash539 1967
[24] Z Odibat and S Momani ldquoNumerical methods for nonlinearpartial differential equations of fractional orderrdquo Applied Math-ematical Modelling vol 32 no 1 pp 28ndash39 2008
[25] A Cuyt and L Wuytack Nonlinear Methods in NumericalAnalysis vol 136 ofNorth-HollandMathematics Studies North-Holland Publishing Amsterdam The Netherlands 1987
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