research article on the inverse problem of the fractional ...lems have many practical applications...
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Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013, Article ID 476154, 8 pageshttp://dx.doi.org/10.1155/2013/476154
Research ArticleOn the Inverse Problem of the Fractional Heat-Like PartialDifferential Equations: Determination of the Source Function
GΓΌlcan Γzkum,1 Ali Demir,1 SertaΓ§ Erman,1 Esra Korkmaz,2 and Berrak ΓzgΓΌr1
1 Department of Mathematics, Science and Letter Faculty, Kocaeli University, Umuttepe Campus, 41380 Kocaeli, Turkey2 Ardahan University, 75000 Ardahan, Turkey
Correspondence should be addressed to GuΜlcan OΜzkum; [email protected]
Received 22 May 2013; Revised 12 September 2013; Accepted 12 September 2013
Academic Editor: H. Srivastava
Copyright Β© 2013 GuΜlcan OΜzkum et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The study in this paper mainly concerns the inverse problem of determining an unknown source function in the linear fractionaldifferential equation with variable coefficient using Adomian decomposition method (ADM). We apply ADM to determine thecontinuous right hand side functions π(π₯) and π(π‘) in the heat-like diffusion equations π·πΌ
π‘π’(π₯, π‘) = β(π₯)π’
π₯π₯(π₯, π‘) + π(π₯) and
π·πΌπ‘π’(π₯, π‘) = β(π₯)π’
π₯π₯(π₯, π‘) + π(π‘), respectively. The results reveal that ADM is very effective and simple for the inverse problem of
determining the source function.
1. Introduction
Fractional differential equations (FDEs) are obtained by gen-eralizing differential equations to an arbitrary order.They areused to model physical systems with memory. Since FDEshave memory, nonlocal relations in space and time complexphenomena can be modeled by using these equations. Dueto this fact, materials with memory and hereditary effects,fluid flow, rheology, diffusive transport, electrical networks,electromagnetic theory and probability, signal processing,and many other physical processes are diverse applicationsof FDEs. Since FDEs are used to model complex phenomena,they play a crucial role in engineering, physics, and appliedmathematics. Therefore, they are generating an increasinginterest from engineers and scientist in the recent years. Asa result, FDEs are quite frequently encountered in differentresearch areas and engineering applications [1].
The book written by Oldham and Spanier [2] played anoutstanding role in the development of the fractional calcu-lus. Also, it was the first book that was entirely devoted to asystematic presentation of the ideas, methods, and applica-tions of the fractional calculus. Afterwards, several funda-mental works on various aspects of the fractional calculusinclude extensive survey on fractional differential equationsby Miller and Ross [3], Podlubny [4], and others. Further,
several references to the books by Oldham and Spanier [2],Miller and Ross [3], and Podlubny [4] show that appliedscientists need first of all an easy introduction to the theoryof fractional derivatives and fractional differential equations,which could help them in their initial steps in adopting thefractional calculus as a method of research [5].
In general, FDEs do not have exact analytical solutions;hence, the approximate and numerical solutions of theseequations are studied [6β8]. Analytical approximations forlinear and nonlinear FDEs are obtained by variational iter-ation method, Adomian decomposition method, homotopyperturbationmethod, Lagrangemultipliermethod, BPs oper-ational matrices method, and so forth. An effective and easy-to-use method for solving such equations is needed. Largeclasses of linear and nonlinear differential equations, bothordinary and partial, can be solved by the Adomian decom-position method [9β12].
Solving an equationwith certain data in a specified regionis called direct problem. On the other hand, determiningan unknown input by using output is called an inverseproblem. This unknown input could be some coefficients,or it could be a source function in equation. Based on thisunknown input the inverse problem is called inverse problemof coefficient identification or inverse problem of sourceidentification, respectively. Generally inverse problems are
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2 Advances in Mathematical Physics
ill-posed problems; that is, they are very sensitive to errorsin measured input. In order to deal with this ill-posedness,regularization methods have been developed. Inverse prob-lems have many practical applications such as geophysics,optics, quantum mechanics, astronomy, medical imagingandmaterials testing, X-ray tomography, and photoelasticity.Theoretical and applied aspects of inverse problems havebeen under intense study lately, especially for the fractionalequation [13β16].
In this paper, we investigate inverse problems of thelinear heat-like differential equations of fractional ordersπ·πΌπ‘π’(π₯, π‘) = β(π₯)π’
π₯π₯(π₯, π‘) + π(π₯) and π·πΌ
π‘π’(π₯, π‘) =
β(π₯)π’π₯π₯(π₯, π‘) + π(π‘) where the function π’(π₯, π‘) is assumed
to be a causal function of time and space. Time fractionalderivative operator π·πΌ
π‘is considered as in Caputo sense [17].
We use the Adomian decompositionmethod [9, 10] to obtainsource functions π(π₯) and π(π‘) under the initial and mixedboundary conditions. By this method, we determine thesource functions π(π₯) and π(π‘) in a rapidly converging seriesform when they exist. Compared with previous researches[10, 12, 18], the method we use in this paper is more effectiveand accurate.
The structure of this paper is given as follows. First, wegive some basic definitions of fractional calculus. Inverseproblem of finding the source function in one-dimensionalfractional heat-like equations with mixed boundary condi-tions is given in Section 2. After that, we give some illustrativeexamples of this method for all cases in Section 3. Finally, theconclusion is given in Section 4.
1.1. Fractional Calculus. In this section, we give basic defini-tions and properties of the fractional calculus [17, 18].
Definition 1. A real function π(π₯), π₯ > 0, is said to be in thespace πΆ
π, π β R if there exists a real number π > π such that
π(π₯) = π₯ππ1(π₯), where π
1(π₯) β πΆ[0,β), and it is said to be
in the space πΆππif π(π) β πΆ
π,π β N.
Definition 2. TheRiemann-Liouville fractional integral oper-ator of order πΌ β₯ 0, of a function π β πΆ
π, π β₯ β1 is defined
as
π½πΌπ (π₯) =1
Ξ (πΌ)β«π₯
0
(π₯ β π‘)πΌβ1π (π‘) ππ‘, πΌ > 0, π₯ > 0
π½0π (π₯) = π (π₯) .
(1)
Some of the basic properties of this operator are given asfollows.
For π β πΆπ, π β₯ β1, πΌ, π½ β₯ 0 and πΎ > β1:
(1) π½πΌπ½π½π (π₯) = π½
πΌ+π½π (π₯) ,
(2) π½πΌπ½π½π (π₯) = π½
π½π½πΌπ (π₯) ,
(3) π½πΌπ₯πΎ =
Ξ (πΎ + 1)
Ξ (πΌ + πΎ + 1)π₯πΌ+πΎ.
(2)
The other properties can be found in [17].
Definition 3. The fractional derivative of π(π₯) in the Caputosense is defined as
π·πΌπ (π₯) = π½πβπΌπ·ππ (π₯)
=1
Ξ (π β πΌ)β«π₯
0
(π₯ β π‘)πβπΌβ1π(π) (π‘) ππ‘,
(3)
whereπ β 1 < πΌ β€ π,π β N, π₯ > 0, π β πΆπβ1.
Useful properties ofπ·πΌ are given as follows.
Lemma 4. If π β 1 < πΌ β€ π, π β N, and π β πΆππ, π β₯ β1,
then
π·πΌπ½πΌπ (π₯) = π (π₯) ,
π½πΌπ·πΌπ (π₯) = π (π₯) βπβ1
βπ=0
π(π) (0+)π₯π
π!, π₯ > 0.
(4)
Since traditional initial and boundary conditions are allowedin problems including Caputo fractional derivatives, it isconsidered here. In this paper, we deal with the fractional heat-like equations where the unknown function π’ = π’(π₯, π‘) is anarbitrary function of time and space.
Definition 5. The Caputo time fractional derivative operatorof order πΌ > 0 is defined as follows where π is the smallestinteger that exceeds πΌ:
π·πΌπ‘π’ (π₯, π‘)
=ππΌπ’ (π₯, π‘)
ππ‘πΌ
=
{{{{{{{{{{{{{
1
Ξ (π β πΌ)
Γβ«π‘
0
(π‘ β π)πβπΌβ1
πππ’ (π₯, π)
ππ‘πππ, for π β 1 < πΌ < π
πππ’ (π₯, π‘)
ππ‘π, for πΌ = π β IN.
(5)
For more details about Caputo fractional differential opera-tor, we refer to [17].
Definition 6. The Mittag-Leffler function with two-parameters is defined by the series expansion as shownbelow, where the real part of πΌ is strictly positive [19]
πΈπΌ,π½
(π§) =β
βπ=0
π§π
Ξ (πΌπ + π½). (6)
2. Inverse Problem ofDetermining Source Function
In this section, we deal with inverse problem of findingthe source function, in one-dimensional fractional heat-likeequationswithmixed boundary conditions. To determine theunknown source function we have developed new methodsthrough ADM as in the following subsections.
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Advances in Mathematical Physics 3
2.1. Determination of Unknown Source Functions Dependingon π₯. We consider the following inverse problem of deter-mining the source function π(π₯):
π·πΌπ‘π’ (π₯, π‘) = β (π₯) π’
π₯π₯(π₯, π‘) + π (π₯) ,
π₯ > 0, π‘ > 0, 0 < πΌ β€ 1,
π’ (π₯, 0) = π1(π₯) ,
π’ (0, π‘) = β1(π‘) ,
π’π₯(0, π‘) = β
2(π‘) ,
(7)
where the functions β1(π‘), β2(π‘) β πΆβ[0,β) and β(π₯), π(π₯),
π1(π₯) β πΆβ[0,β). In order to determine the source function
for this kind of inverse problems, we apply ADM. First,we apply the time-dependent Riemann-Liouville fractionalintegral operator π½πΌ
π‘to both sides of (7) to get rid of fractional
derivativeπ·πΌπ‘as shown below:
π½πΌπ‘π·πΌπ‘π’ (π₯, π‘) = π½
πΌ
π‘(β (π₯) π’
π₯π₯(π₯, π‘)) + π½
πΌ
π‘π (π₯) . (8)
Then we get
π’ (π₯, π‘) = π’ (π₯, 0) + π½πΌ
π‘π (π₯) + π½
πΌ
π‘(β (π₯) π’
π₯π₯(π₯, π‘)) . (9)
In ADM the solution π’(π₯, π‘) is written in the following seriesform [9]:
π’ (π₯, π‘) =β
βπ=0
π’π(π₯, π‘) , (10)
where π’ and π’π, π β N, are defined in πΆβ[0,β) Γ πΆ1
π[0,β).
After substituting the decomposition (10) into (9) and settingthe recurrence scheme as follows:
π’0(π₯, π‘) = π’ (π₯, 0) + π½
πΌ
π‘π (π₯) ,
π’π+1
(π₯, π‘) = π½πΌ
π‘(β (π₯) (π’
π)π₯π₯
(π₯, π‘)) , π = 0, 1, . . . ,(11)
we get ADM polynomials below
π’0(π₯, π‘) = π
1(π₯) + π (π₯)
π‘πΌ
Ξ (πΌ + 1),
π’1(π₯, π‘) = π½
πΌ
π‘(β (π₯) (π’
0)π₯π₯
(π₯))
= β (π₯) π
1(π₯)
π‘πΌ
Ξ (πΌ + 1)
+ β (π₯) π
(π₯)π‘2πΌ
Ξ (2πΌ + 1),
π’2(π₯, π‘)
= π½πΌπ‘(β (π₯) (π’
1)π₯π₯
(π₯))
= β (π₯) {[β
(π₯) π
1(π₯) + β
(π₯) π
1(π₯)
+ β (π₯) π
1(π₯) + β (π₯) π
(πV)1
(π₯)]π‘2πΌ
Ξ (2πΌ + 1)
+ [β (π₯) π
(π₯) + 2β
(π₯) π
(π₯)
+ β (π₯) π(πV)
(π₯)]π‘3πΌ
Ξ (3πΌ + 1)} ,
...(12)
After writing these polynomials in (10), the solution π’(π₯, π‘) isgiven by
π’ (π₯, π‘)
= π1(π₯) + π (π₯)
π‘πΌ
Ξ (πΌ + 1)+ β (π₯) π
1(π₯)
π‘πΌ
Ξ (πΌ + 1)
+ β (π₯) π
(π₯)π‘2πΌ
Ξ (2πΌ + 1)
+ β (π₯) {[β
(π₯) π
1(π₯) + β
(π₯) π
1(π₯)
+ β (π₯) π
1(π₯) + β (π₯) π
(πV)1
(π₯)]π‘2πΌ
Ξ (2πΌ + 1)
+ [β (π₯) π
(π₯) + 2β
(π₯) π
(π₯)
+ β (π₯) π(πV)
(π₯)]π‘3πΌ
Ξ (3πΌ + 1)} + β β β .
(13)
If we arrange it with respect to like powers of π‘, then we get
π’ (π₯, π‘)
= π1(π₯) + [π (π₯) + β (π₯) π
1(π₯)]
π‘πΌ
Ξ (πΌ + 1)
+ β (π₯) {[π
(π₯) + β
(π₯) π
1(π₯) + β
(π₯) π
1(π₯)
+ β (π₯) π
1(π₯) + β (π₯) π
(πV)1
(π₯)]π‘2πΌ
Ξ (2πΌ + 1)
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4 Advances in Mathematical Physics
+ [β (π₯) π
(π₯) + 2β
(π₯) π
(π₯)
+ β (π₯) π(πV)
(π₯)]π‘3πΌ
Ξ (3πΌ + 1)} + β β β .
(14)
To determine the unknown source function, first we expandthe boundary conditions π’(0, π‘) = β
1(π‘) and π’
π₯(0, π‘) =
β2(π‘) into the following series for the space whose bases are
{π‘ππΌ/Ξ(ππΌ + 1)}β
π=0, 0 < πΌ β€ 1:
β1(π‘) = β
1(0) + β
1(0)
π‘πΌ
Ξ (πΌ + 1)+ β1(0)
π‘2πΌ
Ξ (2πΌ + 1)+ β β β ,
(15)
β2(π‘) = β
2(0) + β
2(0)
π‘πΌ
Ξ (πΌ + 1)+ β2(0)
π‘2πΌ
Ξ (2πΌ + 1)+ β β β .
(16)
On the other hand, if we rewrite the boundary conditionsπ’(0, π‘) and π’
π₯(0, π‘) from (14), then we have
β1(π‘) = π
1(0) + [π (0) + β (0) π
1(0)]
π‘πΌ
Ξ (πΌ + 1)
+ β (0) {[π
(0) + β
(0) π
1(0) + β
(0) π
1(0)
+ β (0) π
1(0)+ β (0) π
(πV)1
(0)]π‘2πΌ
Ξ (2πΌ + 1)
+ [β (0) π
(0) + 2β
(0) π
(0)
+ β (0) π(πV)
(0)]π‘3πΌ
Ξ (3πΌ + 1)} + β β β ,
(17)
β2(π‘) = π
1(0) + [π
(0) + β
(0) π
1(0)
+ β (0) π
1(0)]
π‘πΌ
Ξ (πΌ + 1)
+ {β (0) [π
(0) + β
(0) π
1(0) + β
(0) π
1(0)
+ β (0) π
1(0) + β (0) π
(πV)1
(0)]
+ β (0) [π
(0) + β
(0) π
1(0)
+ 2β (0) π
1(0)
+ β (0) π
1(0) + β
(0) π
1(0)
+ 2β (0) π(πV)1
(0)
+ β (0) π(V)1
(0)]}π‘2πΌ
Ξ (2πΌ + 1)
+ {β (0) [β
(0) π
(0) + 2β
(0) π
(0)
+ β (0) π(πV)
(0) ]
+ β (0) [β
(0) π
(0) + 3β
(0) π
(0)
+ 3β (0) π(πV)
(0)
+ β (0) π(V)
(0) ]}π‘3πΌ
Ξ (3πΌ + 1)+ β β β .
(18)
Equating (15) and (17) yields the following:
β1(0) = π
1(0) ,
β1(0) = π (0) + β (0) π
1(0) ,
β1(0) = β (0) [π
(0) + β
(0) π
1(0) + β
(0) π
1(0)
+ β (0) π
1(0) + β (0) π
(πV)1
(0)] ,
...
(19)
and equating (16) and (18) yields the following:
β2(0) = π
1(0) ,
β2(0) = π
(0) + β
(0) π
1(0) + β (0) π
1(0) ,
(20)
β2(0) = β
(0) [π
(0) + β
(0) π
1(0) + β
(0) π
1(0)
+β (0) π
1(0) + β (0) π
(πV)1
(0)]
+ β (0) [π
(0) + β
(0) π
1(0) + 2β
(0) π
1(0)
+ β (0) π
1(0) + β
(0) π
1(0)
+2β (0) π(πV)1
(0) + β (0) π(V)1
(0)] ,
...(21)
Using the above data in the following Taylor series expansionof unknown function π(π₯) we get
π (π₯) = π (0) + π
(0) π₯ + π
(0)π₯2
2!+ π (0)
π₯3
3!+ β β β .
(22)
Consequently, we determine π(π₯) as follows:
π (π₯) = [β
1(0) β β (0) π
1(0)]
+ [β2(0) β β
(0) π
1(0) β β (0) π
1(0)] π₯
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Advances in Mathematical Physics 5
+ [β1(0)
β (0)β β (0) π
1(0) β β
(0) π
1(0)
β β (0) π
1(0) β β (0) π
(πV)1
(0)]π₯2
2!+ β β β ,
(23)
where β(0) ΜΈ= 0.
2.2. Determination of Unknown Source Functions Dependingon π‘. Weconsider the following inverse problemof determin-ing the source function π(π‘):
π·πΌπ‘π’ (π₯, π‘) = β (π₯) π’
π₯π₯(π₯, π‘) + π (π‘) ,
π₯ > 0, π‘ > 0, 0 < πΌ β€ 1,
π’ (π₯, 0) = π1(π₯) ,
π’ (0, π‘) = β1(π‘) ,
π’π₯(0, π‘) = β
2(π‘) ,
(24)
where β1(π‘), β2(π‘) β πΆβ[0,β), β(π₯), π
1(π₯) β πΆβ[0,β), and
π(π‘) β πΆ1π[0,β), π β₯ β1. As in the previous case, we apply
ADM to determine the unknown function π(π‘).First, to reduce the problem, we define new functions in
the following form:
π€ (π‘) = π½πΌ
π‘π (π‘)
π’ (π₯, π‘) = V (π₯, π‘) + π€ (π‘) .(25)
Then the reduced problem is given as follows:
π·πΌπ‘V (π₯, π‘) = β (π₯) V
π₯π₯(π₯, π‘) , (26)
with the following initial and mixed boundary conditions
V (π₯, 0) = π1(π₯) β π€ (0) ,
V (0, π‘) = β1(π‘) β π€ (π‘) ,
Vπ₯(0, π‘) = β
2(π‘) .
(27)
By using ADM as in the previous section, we determine thefunction π€(π‘) which leads to the source function π(π‘). Let usapply π½πΌ
π‘to both sides of (26) as shown below
π½πΌπ‘π·πΌπ‘V (π₯, π‘) = π½πΌ
π‘(β (π₯) V
π₯π₯(π₯, π‘)) . (28)
Then we get
V (π₯, π‘) = V (π₯, 0) + π½πΌπ‘(β (π₯) V
π₯π₯(π₯, π‘)) . (29)
Now we define the solution V(π₯, π‘) by the following decom-position series according to ADM
V (π₯, π‘) =β
βπ=0
Vπ(π₯, π‘) . (30)
Substituting (30) into (29), we obtain
V0(π₯, π‘) = V (π₯, 0) ,
Vπ+1
(π₯, π‘) = π½πΌ
π‘(β (π₯) (V
π)π₯π₯
(π₯, π‘)) , π = 0, 1, . . . .(31)
Hence, the recurrence scheme is obtained as follows:V0(π₯, π‘) = V (π₯, 0) = π
1(π₯) β π€ (0) ,
V1(π₯, π‘) = π½
πΌ
π‘(β (π₯) (V
0)π₯π₯
(π₯)) = β (π₯) π
1(π₯)
π‘πΌ
Ξ (πΌ + 1),
V2(π₯, π‘) = π½
πΌ
π‘(β (π₯) (V
1)π₯π₯
(π₯)) = β2
(π₯) π(πV)1
(π₯)π‘2πΌ
Ξ (2πΌ + 1),
...(32)
Consequently, from (30), the solution V(π₯, π‘) is given as shownbelow
V (π₯, π‘) = V0(π₯, π‘) + V
1(π₯, π‘) + V
2(π₯, π‘) + β β β
= π1(π₯) β π€ (0) + β (π₯) π
1(π₯)
π‘πΌ
Ξ (πΌ + 1)
+ β2 (π₯) π(πV)1
(π₯)π‘2πΌ
Ξ (2πΌ + 1)+ β β β .
(33)
By using the boundary condition V(0, π‘) = β1(π‘) + π€(π‘) and
π€(0) = 0, we have
π€ (π‘) = π1(0) β β
1(π‘) + β (0) π
1(0)
π‘πΌ
Ξ (πΌ + 1)
+ β2 (0) π(πV)1
(0)π‘2πΌ
Ξ (2πΌ + 1)+ β β β ,
(34)
which implies the following:
π½πΌπ‘π (π‘) = π
1(0) β β
1(π‘) + β (0) π
1(0)
π‘πΌ
Ξ (πΌ + 1)
+ β2 (0) π(πV)1
(0)π‘2πΌ
Ξ (2πΌ + 1)+ β β β .
(35)
Since π·πΌπ‘π€(π₯, π‘) = π·πΌ
π‘π½πΌπ‘π(π‘) = π(π‘), we obtain the source
function π(π‘) as follows:
π (π‘) = π·πΌ
π‘[π1(π₯) β β
1(π‘) + β (π₯) π
1(π₯)
π‘πΌ
Ξ (πΌ + 1)
+ β2 (π₯) π(πV)1
(π₯)π‘2πΌ
Ξ (2πΌ + 1)+ β β β ] .
(36)
3. Examples
Example 1. We consider the inverse problem of determiningsource function π(π₯) in the following one-dimensional frac-tional heat-like PDE:
π·πΌπ‘π’ (π₯, π‘) = 2π’
π₯π₯(π₯, π‘) + π (π₯) , π₯ > 0, 0 < πΌ β€ 1, π‘ > 0,
(37)
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6 Advances in Mathematical Physics
subject to the following initial and nonhomogeneous mixedboundary conditions:
π’ (π₯, 0) = ππ₯ + sinπ₯,
π’ (0, π‘) = π2π‘,
π’π₯(0, π‘) = π
2π‘+1.
(38)
Now, let us apply the time-dependent Riemann Liouvillefractional integral operator π½πΌ
π‘to both sides of (37)
π½πΌπ‘π·πΌπ‘π’ (π₯, π‘) = 2π½
πΌ
π‘π’π₯π₯
(π₯, π‘) + π½πΌ
π‘π (π₯) (39)
which implies
π’ (π₯, π‘) β π’ (π₯, 0) = 2π½πΌ
π‘π’π₯π₯
(π₯, π‘) + π (π₯) π½πΌ
π‘(1) . (40)
Then, from the initial condition we get
π’ (π₯, π‘) = ππ₯ + sinπ₯ + π (π₯) π‘
πΌ
Ξ (πΌ + 1)+ 2π½πΌπ‘π’π₯π₯
(π₯, π‘) . (41)
Now, we apply ADM to the problem. In (41), the sum of thefirst three terms is identified as π’
0. So
π’0= ππ₯ + sinπ₯ + π (π₯) π‘
πΌ
Ξ (πΌ + 1),
π’π+1
= 2π½πΌπ‘(π’π)π₯π₯
(π₯, π‘) , π β₯ 0.
(42)
For π = 0, we have
π’1= 2π½πΌπ‘(π’0)π₯π₯
(π₯, π‘)
= 2ππ₯π‘πΌ
Ξ (πΌ + 1)β 2 sinπ₯ π‘
πΌ
Ξ (πΌ + 1)
+ 2π (π₯)π‘2πΌ
Ξ (2πΌ + 1)+ β β β ,
(43)
similarly, for π = 1, we have
π’2= 2π½πΌπ‘(π’1)π₯π₯
(π₯, π‘)
= 4ππ₯π‘2πΌ
Ξ (2πΌ + 1)+ 4 sinπ₯ π‘
2πΌ
Ξ (2πΌ + 1)
+ 4π(πV) (π₯)π‘3πΌ
Ξ (3πΌ + 1)+ β β β ,
(44)
and for π = 2, we have
π’3=2π½πΌπ‘(π’2)π₯π₯
(π₯, π‘)
= 8ππ₯π‘3πΌ
Ξ (3πΌ + 1)β 8 sinπ₯ π‘
3πΌ
Ξ (3πΌ + 1)
+ 8π(Vπ) (π₯)π‘4πΌ
Ξ (4πΌ + 1)+ β β β
...
(45)
Then using ADM polynomials, we get the solution π’(π₯, π‘) asfollows:π’ (π₯, π‘) = π’
0+ π’1+ π’2+ π’3+ β β β
= ππ₯ + sinπ₯ + π (π₯) π‘πΌ
Ξ (πΌ + 1)+ 2ππ₯
π‘πΌ
Ξ (πΌ + 1)
β 2 sinπ₯ π‘πΌ
Ξ (πΌ + 1)+ 2π (π₯)
π‘2πΌ
Ξ (2πΌ + 1)
+ 4ππ₯π‘2πΌ
Ξ (2πΌ + 1)+ 4 sinπ₯ π‘
2πΌ
Ξ (2πΌ + 1)
+ 4π(πV) (π₯)π‘3πΌ
Ξ (3πΌ + 1)+ 8ππ₯
π‘3πΌ
Ξ (3πΌ + 1)
β 8 sinπ₯ π‘3πΌ
Ξ (3πΌ + 1)+ 8π(Vπ) (π₯)
π‘4πΌ
Ξ (4πΌ + 1)+ β β β .
(46)
After arranging it according to like powers of π‘, we have
π’ (π₯, π‘) = ππ₯ + sinπ₯ + π‘
πΌ
Ξ (πΌ + 1)[π (π₯) + 2π
π₯ β 2 sinπ₯]
+π‘2πΌ
Ξ (2πΌ + 1)[2π (π₯) + 4π
π₯ + 4 sinπ₯]
+π‘3πΌ
Ξ (3πΌ + 1)[4π(πV) (π₯) + 8π
π₯ β 8 sinπ₯]
+π‘4πΌ
Ξ (4πΌ + 1)[8π(Vπ) (π₯) + 16π
π₯ + 16 sinπ₯] + β β β .
(47)
Now, by applying the boundary condition given in (38), weobtain
π’ (0, π‘) = 1 +π‘πΌ
Ξ (πΌ + 1)[π (0) + 2] +
π‘2πΌ
Ξ (2πΌ + 1)[2π (0) + 4]
+π‘3πΌ
Ξ (3πΌ + 1)[4π(πV) (0) + 8]
+π‘4πΌ
Ξ (4πΌ + 1)[8π(Vπ) (0) + 16] + β β β .
(48)
From (15), it must be equal to the following Taylorseries expansion of π2π‘ in the space whose bases are{π‘ππΌ/Ξ(ππΌ + 1)}
β
π=0, 0 < πΌ β€ 1:
π2π‘ = 1 + 2π‘πΌ
Ξ (πΌ + 1)+ 4
π‘2πΌ
Ξ (2πΌ + 1)
+ 8π‘3πΌ
Ξ (3πΌ + 1)+ 16
π‘4πΌ
Ξ (4πΌ + 1)+ β β β .
(49)
Hence, from the equality of the coefficients of correspondingterms, we get
π (0) = π
(0) = π(πV)
(0) = π(Vπ)
(0) = β β β = 0. (50)
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Advances in Mathematical Physics 7
From (47), we have
π’π₯(π₯, π‘) = π
π₯ + cosπ₯ + π‘πΌ
Ξ (πΌ + 1)[π (π₯) + 2π
π₯ β 2 cosπ₯]
+π‘2πΌ
Ξ (2πΌ + 1)[2π (π₯) + 4π
π₯ + 4 cosπ₯]
+π‘3πΌ
Ξ (3πΌ + 1)[4π(V) (π₯) + 8π
π₯ β 8 cosπ₯]
+π‘4πΌ
Ξ (4πΌ + 1)[8π(Vππ) (π₯) + 16π
π₯ + 16 cosπ₯] + β β β .
(51)
So,
π’π₯(0, π‘) = 2 +
π‘πΌ
Ξ (πΌ + 1)π (0) +
π‘2πΌ
Ξ (2πΌ + 1)[2π (0) + 8]
+π‘3πΌ
Ξ (3πΌ + 1)[4π(V) (0)]
+π‘4πΌ
Ξ (4πΌ + 1)[8π(Vππ) (0) + 32] + β β β .
(52)
From the derivative boundary condition given in (38), it mustbe equal to the following series expansion of π2π‘+1 in the spacewhose bases are {π‘ππΌ/Ξ(ππΌ + 1)}β
π=0, 0 < πΌ β€ 1:
π2π‘ + 1 = 2 + 2π‘πΌ
Ξ (πΌ + 1)+ 4
π‘2πΌ
Ξ (2πΌ + 1)
+ 8π‘3πΌ
Ξ (3πΌ + 1)+ 16
π‘4πΌ
Ξ (4πΌ + 1)+ β β β .
(53)
Then, we find the following data:
π (0) = 2, π
(0) = β2, π(V)
(0) = 2,
π(Vππ) (0) = β2, . . . .(54)
Next, using (50) and (54), we have the Taylor series expansionof π(π₯) as follows:
π (π₯) = π (0) + π
(0) π₯ + π
(0)π₯2
2!
+ π (0)π₯3
3!+ π(πV) (0)
π₯4
4!+ β β β
= 2π₯ β 2π₯3
3!+ 2
π₯5
5!+ 2
π₯7
7!+ β β β .
(55)
That is,
π (π₯) = 2 [π₯ βπ₯3
3!+π₯5
5!+π₯7
7!+ β β β ] (56)
which is the series expansion of the function 2 sinπ₯. Conse-quently, we determine the source function π(π₯) as
π (π₯) = 2 sinπ₯. (57)
Example 2. We consider the inverse problem of determiningsource function π(π‘) in the following one-dimensional frac-tional heat-like diffusion equation:
π·πΌπ‘π’ (π₯, π‘) =
1
2π₯2π’π₯π₯
(π₯, π‘) + π (π‘) ,
π₯ > 0, 0 < πΌ β€ 1, π‘ > 0,
(58)
subject to following initial and mixed boundary conditions
π’ (π₯, 0) = π₯2 +
1
2, π’ (0, π‘) =
π2π‘
2, π’
π₯(0, π‘) = 0.
(59)
Now let us determine the source function π(π‘). To reduce theproblem, we define new functions as follows:
π€ (π‘) = π½πΌ
π‘π (π‘) ,
π’ (π₯, π‘) = V (π₯, π‘) + π€ (π‘) .(60)
Then, our reduced problem is given as follows:
π·πΌπ‘V (π₯, π‘) =
1
2π₯2Vπ₯π₯
(π₯, π‘) , 0 < πΌ β€ 1, π‘ > 0
V (π₯, 0) = π’ (π₯, 0) β π€ (0) = π₯2 +1
2,
V (0, π‘) = π’ (0, π‘) β π€ (π‘) =π2π‘
2β π€ (π‘) ,
Vπ₯(0, π‘) = 0.
(61)
Applying π½πΌπ‘to both sides of (61), then we get
V (π₯, π‘) β V (π₯, 0) =1
2π₯2π½πΌπ‘Vπ₯π₯
(π₯, π‘) (62)
which implies
V (π₯, π‘) = π₯2 +1
2+1
2π₯2π½πΌπ‘Vπ₯π₯
(π₯, π‘) . (63)
By using ADM for (63), we obtain
V0= π₯2 +
1
2,
Vπ+1
=1
2π₯2π½πΌπ‘(Vπ)π₯π₯
(π₯, π‘) , π β₯ 0.
(64)
Then, for π = 0, we get
V1=
1
2π₯2π½πΌπ‘(V0)π₯π₯
(π₯, π‘)
= π₯2π‘πΌ
Ξ (πΌ + 1),
(65)
similarly, for π = 1, we get
V2=
1
2π₯2π½πΌπ‘(V1)π₯π₯
(π₯, π‘)
= π₯2π‘2πΌ
Ξ (2πΌ + 1),
(66)
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8 Advances in Mathematical Physics
and for π = 2, we get
V3=
1
2π₯2π½πΌπ‘(V2)π₯π₯
(π₯, π‘)
= π₯2π‘3πΌ
Ξ (3πΌ + 1),
...
(67)
As a result, we get the solution V as follows:
V (π₯, π‘) = V0+ V1+ V2+ V3+ β β β
= π₯2 +1
2+ π₯2
π‘πΌ
Ξ (πΌ + 1)
+ π₯2π‘2πΌ
Ξ (2πΌ + 1)+ π₯2
π‘3πΌ
Ξ (3πΌ + 1)+ β β β
=1
2+ π₯2 [1 +
π‘πΌ
Ξ (πΌ + 1)+
π‘2πΌ
Ξ (2πΌ + 1)
+π‘3πΌ
Ξ (3πΌ + 1)+ β β β ] .
(68)
Therefore, from the boundary condition we have
π€ (π‘) =π2π‘
2β1
2. (69)
Using (69) in the definition π·πΌπ‘π€(π‘) = π·πΌ
π‘π½πΌπ‘π(π‘) =
π(π‘), finally we obtain the source function π(π‘) as π(π‘) =π·πΌπ‘((π2π‘/2) β (1/2)); that is,
π (π‘) =1
2π·πΌπ‘(π2π‘) . (70)
Here,
π·πΌπ‘(π2π‘) = π‘βπΌπΈ
1,1βπΌ(2π‘) , (71)
where πΈ1,1βπΌ
is Mittag-Leffler function with two parametersgiven as; (6).
4. Conclusion
Thebest part of this method is that one can easily apply ADMto the fractional partial differential equations like applyingADM to ordinary differential equations.
References
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[2] K. B. Oldham and J. Spanier,The Fractional Calculus, AcademicPress, New York, NY, USA, 1974.
[3] K. S.Miller and B. Ross,An Introduction to the Fractional Calcu-lus and Fractional Differential Equations, A Wiley-IntersciencePublication, John Wiley & Sons, New York, NY, USA, 1993.
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[8] M. Alipour and D. Baleanu, βApproximate analytical solutionfor nonlinear system of fractional differential equations by BPsoperational matrices,β Advances in Mathematical Physics, vol.2013, Article ID 954015, 9 pages, 2013.
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