research article on the inverse problem of the fractional ...lems have many practical applications...

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

Gülcan Özkum,1 Ali Demir,1 Sertaç Erman,1 Esra Korkmaz,2 and Berrak Özgür1

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 Gülcan Özkum; [email protected]

Received 22 May 2013; Revised 12 September 2013; Accepted 12 September 2013

Academic Editor: H. Srivastava

Copyright © 2013 Gülcan Özkum 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 𝑓(𝑥) and 𝑓(𝑡) in the heat-like diffusion equations 𝐷𝛼

𝑡𝑢(𝑥, 𝑡) = ℎ(𝑥)𝑢

𝑥𝑥(𝑥, 𝑡) + 𝑓(𝑥) and

𝐷𝛼𝑡𝑢(𝑥, 𝑡) = ℎ(𝑥)𝑢

𝑥𝑥(𝑥, 𝑡) + 𝑓(𝑡), 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 [6–8]. 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 [9–12].

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

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 [13–16].

In this paper, we investigate inverse problems of thelinear heat-like differential equations of fractional orders𝐷𝛼𝑡𝑢(𝑥, 𝑡) = ℎ(𝑥)𝑢

𝑥𝑥(𝑥, 𝑡) + 𝑓(𝑥) and 𝐷𝛼

𝑡𝑢(𝑥, 𝑡) =

ℎ(𝑥)𝑢𝑥𝑥(𝑥, 𝑡) + 𝑓(𝑡) where the function 𝑢(𝑥, 𝑡) is assumed

to be a causal function of time and space. Time fractionalderivative operator 𝐷𝛼

𝑡is considered as in Caputo sense [17].

We use the Adomian decompositionmethod [9, 10] to obtainsource functions 𝑓(𝑥) and 𝑓(𝑡) under the initial and mixedboundary conditions. By this method, we determine thesource functions 𝑓(𝑥) and 𝑓(𝑡) 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 𝑓(𝑥), 𝑥 > 0, is said to be in thespace 𝐶

𝜇, 𝜇 ∈ R if there exists a real number 𝑝 > 𝜇 such that

𝑓(𝑥) = 𝑥𝑝𝑓1(𝑥), where 𝑓

1(𝑥) ∈ 𝐶[0,∞), and it is said to be

in the space 𝐶𝑚𝜇if 𝑓(𝑚) ∈ 𝐶

𝜇,𝑚 ∈ N.

Definition 2. TheRiemann-Liouville fractional integral oper-ator of order 𝛼 ≥ 0, of a function 𝑓 ∈ 𝐶

𝜇, 𝜇 ≥ −1 is defined

as

𝐽𝛼𝑓 (𝑥) =1

Γ (𝛼)∫𝑥

0

(𝑥 − 𝑡)𝛼−1𝑓 (𝑡) 𝑑𝑡, 𝛼 > 0, 𝑥 > 0

𝐽0𝑓 (𝑥) = 𝑓 (𝑥) .

(1)

Some of the basic properties of this operator are given asfollows.

For 𝑓 ∈ 𝐶𝜇, 𝜇 ≥ −1, 𝛼, 𝛽 ≥ 0 and 𝛾 > −1:

(1) 𝐽𝛼𝐽𝛽𝑓 (𝑥) = 𝐽

𝛼+𝛽𝑓 (𝑥) ,

(2) 𝐽𝛼𝐽𝛽𝑓 (𝑥) = 𝐽

𝛽𝐽𝛼𝑓 (𝑥) ,

(3) 𝐽𝛼𝑥𝛾 =

Γ (𝛾 + 1)

Γ (𝛼 + 𝛾 + 1)𝑥𝛼+𝛾.

(2)

The other properties can be found in [17].

Definition 3. The fractional derivative of 𝑓(𝑥) in the Caputosense is defined as

𝐷𝛼𝑓 (𝑥) = 𝐽𝑚−𝛼𝐷𝑚𝑓 (𝑥)

=1

Γ (𝑚 − 𝛼)∫𝑥

0

(𝑥 − 𝑡)𝑚−𝛼−1𝑓(𝑚) (𝑡) 𝑑𝑡,

(3)

where𝑚 − 1 < 𝛼 ≤ 𝑚,𝑚 ∈ N, 𝑥 > 0, 𝑓 ∈ 𝐶𝑚−1.

Useful properties of𝐷𝛼 are given as follows.

Lemma 4. If 𝑚 − 1 < 𝛼 ≤ 𝑚, 𝑚 ∈ N, and 𝑓 ∈ 𝐶𝑚𝜇, 𝜇 ≥ −1,

then

𝐷𝛼𝐽𝛼𝑓 (𝑥) = 𝑓 (𝑥) ,

𝐽𝛼𝐷𝛼𝑓 (𝑥) = 𝑓 (𝑥) −𝑚−1

∑𝑘=0

𝑓(𝑘) (0+)𝑥𝑘

𝑘!, 𝑥 > 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 𝑢 = 𝑢(𝑥, 𝑡) is anarbitrary function of time and space.

Definition 5. The Caputo time fractional derivative operatorof order 𝛼 > 0 is defined as follows where 𝑚 is the smallestinteger that exceeds 𝛼:

𝐷𝛼𝑡𝑢 (𝑥, 𝑡)

=𝜕𝛼𝑢 (𝑥, 𝑡)

𝜕𝑡𝛼

=

{{{{{{{{{{{{{

1

Γ (𝑚 − 𝛼)

×∫𝑡

0

(𝑡 − 𝜏)𝑚−𝛼−1

𝜕𝑚𝑢 (𝑥, 𝜏)

𝜕𝑡𝑚𝑑𝜏, for 𝑚 − 1 < 𝛼 < 𝑚

𝜕𝑚𝑢 (𝑥, 𝑡)

𝜕𝑡𝑚, for 𝛼 = 𝑚 ∈ 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 𝛼 is strictly positive [19]

𝐸𝛼,𝛽

(𝑧) =∞

∑𝑛=0

𝑧𝑛

Γ (𝛼𝑛 + 𝛽). (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.

Advances in Mathematical Physics 3

2.1. Determination of Unknown Source Functions Dependingon 𝑥. We consider the following inverse problem of deter-mining the source function 𝑓(𝑥):

𝐷𝛼𝑡𝑢 (𝑥, 𝑡) = ℎ (𝑥) 𝑢

𝑥𝑥(𝑥, 𝑡) + 𝑓 (𝑥) ,

𝑥 > 0, 𝑡 > 0, 0 < 𝛼 ≤ 1,

𝑢 (𝑥, 0) = 𝑓1(𝑥) ,

𝑢 (0, 𝑡) = ℎ1(𝑡) ,

𝑢𝑥(0, 𝑡) = ℎ

2(𝑡) ,

(7)

where the functions ℎ1(𝑡), ℎ2(𝑡) ∈ 𝐶∞[0,∞) and ℎ(𝑥), 𝑓(𝑥),

𝑓1(𝑥) ∈ 𝐶∞[0,∞). 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 𝐽𝛼

𝑡to both sides of (7) to get rid of fractional

derivative𝐷𝛼𝑡as shown below:

𝐽𝛼𝑡𝐷𝛼𝑡𝑢 (𝑥, 𝑡) = 𝐽

𝛼

𝑡(ℎ (𝑥) 𝑢

𝑥𝑥(𝑥, 𝑡)) + 𝐽

𝛼

𝑡𝑓 (𝑥) . (8)

Then we get

𝑢 (𝑥, 𝑡) = 𝑢 (𝑥, 0) + 𝐽𝛼

𝑡𝑓 (𝑥) + 𝐽

𝛼

𝑡(ℎ (𝑥) 𝑢

𝑥𝑥(𝑥, 𝑡)) . (9)

In ADM the solution 𝑢(𝑥, 𝑡) is written in the following seriesform [9]:

𝑢 (𝑥, 𝑡) =∞

∑𝑛=0

𝑢𝑛(𝑥, 𝑡) , (10)

where 𝑢 and 𝑢𝑛, 𝑛 ∈ N, are defined in 𝐶∞[0,∞) × 𝐶1

𝜇[0,∞).

After substituting the decomposition (10) into (9) and settingthe recurrence scheme as follows:

𝑢0(𝑥, 𝑡) = 𝑢 (𝑥, 0) + 𝐽

𝛼

𝑡𝑓 (𝑥) ,

𝑢𝑛+1

(𝑥, 𝑡) = 𝐽𝛼

𝑡(ℎ (𝑥) (𝑢

𝑛)𝑥𝑥

(𝑥, 𝑡)) , 𝑛 = 0, 1, . . . ,(11)

we get ADM polynomials below

𝑢0(𝑥, 𝑡) = 𝑓

1(𝑥) + 𝑓 (𝑥)

𝑡𝛼

Γ (𝛼 + 1),

𝑢1(𝑥, 𝑡) = 𝐽

𝛼

𝑡(ℎ (𝑥) (𝑢

0)𝑥𝑥

(𝑥))

= ℎ (𝑥) 𝑓

1(𝑥)

𝑡𝛼

Γ (𝛼 + 1)

+ ℎ (𝑥) 𝑓

(𝑥)𝑡2𝛼

Γ (2𝛼 + 1),

𝑢2(𝑥, 𝑡)

= 𝐽𝛼𝑡(ℎ (𝑥) (𝑢

1)𝑥𝑥

(𝑥))

= ℎ (𝑥) {[ℎ

(𝑥) 𝑓

1(𝑥) + ℎ

(𝑥) 𝑓

1(𝑥)

+ ℎ (𝑥) 𝑓

1(𝑥) + ℎ (𝑥) 𝑓

(𝑖V)1

(𝑥)]𝑡2𝛼

Γ (2𝛼 + 1)

+ [ℎ (𝑥) 𝑓

(𝑥) + 2ℎ

(𝑥) 𝑓

(𝑥)

+ ℎ (𝑥) 𝑓(𝑖V)

(𝑥)]𝑡3𝛼

Γ (3𝛼 + 1)} ,

...(12)

After writing these polynomials in (10), the solution 𝑢(𝑥, 𝑡) isgiven by

𝑢 (𝑥, 𝑡)

= 𝑓1(𝑥) + 𝑓 (𝑥)

𝑡𝛼

Γ (𝛼 + 1)+ ℎ (𝑥) 𝑓

1(𝑥)

𝑡𝛼

Γ (𝛼 + 1)

+ ℎ (𝑥) 𝑓

(𝑥)𝑡2𝛼

Γ (2𝛼 + 1)

+ ℎ (𝑥) {[ℎ

(𝑥) 𝑓

1(𝑥) + ℎ

(𝑥) 𝑓

1(𝑥)

+ ℎ (𝑥) 𝑓

1(𝑥) + ℎ (𝑥) 𝑓

(𝑖V)1

(𝑥)]𝑡2𝛼

Γ (2𝛼 + 1)

+ [ℎ (𝑥) 𝑓

(𝑥) + 2ℎ

(𝑥) 𝑓

(𝑥)

+ ℎ (𝑥) 𝑓(𝑖V)

(𝑥)]𝑡3𝛼

Γ (3𝛼 + 1)} + ⋅ ⋅ ⋅ .

(13)

If we arrange it with respect to like powers of 𝑡, then we get

𝑢 (𝑥, 𝑡)

= 𝑓1(𝑥) + [𝑓 (𝑥) + ℎ (𝑥) 𝑓

1(𝑥)]

𝑡𝛼

Γ (𝛼 + 1)

+ ℎ (𝑥) {[𝑓

(𝑥) + ℎ

(𝑥) 𝑓

1(𝑥) + ℎ

(𝑥) 𝑓

1(𝑥)

+ ℎ (𝑥) 𝑓

1(𝑥) + ℎ (𝑥) 𝑓

(𝑖V)1

(𝑥)]𝑡2𝛼

Γ (2𝛼 + 1)


+ [ℎ (𝑥) 𝑓

(𝑥) + 2ℎ

(𝑥) 𝑓

(𝑥)

+ ℎ (𝑥) 𝑓(𝑖V)

(𝑥)]𝑡3𝛼

Γ (3𝛼 + 1)} + ⋅ ⋅ ⋅ .

(14)

To determine the unknown source function, first we expandthe boundary conditions 𝑢(0, 𝑡) = ℎ

1(𝑡) and 𝑢

𝑥(0, 𝑡) =

ℎ2(𝑡) into the following series for the space whose bases are

{𝑡𝑛𝛼/Γ(𝑛𝛼 + 1)}∞

𝑛=0, 0 < 𝛼 ≤ 1:

ℎ1(𝑡) = ℎ

1(0) + ℎ

1(0)

𝑡𝛼

Γ (𝛼 + 1)+ ℎ1(0)

𝑡2𝛼

Γ (2𝛼 + 1)+ ⋅ ⋅ ⋅ ,

(15)

ℎ2(𝑡) = ℎ

2(0) + ℎ

2(0)

𝑡𝛼

Γ (𝛼 + 1)+ ℎ2(0)

𝑡2𝛼

Γ (2𝛼 + 1)+ ⋅ ⋅ ⋅ .

(16)

On the other hand, if we rewrite the boundary conditions𝑢(0, 𝑡) and 𝑢

𝑥(0, 𝑡) from (14), then we have

ℎ1(𝑡) = 𝑓

1(0) + [𝑓 (0) + ℎ (0) 𝑓

1(0)]

𝑡𝛼

Γ (𝛼 + 1)

+ ℎ (0) {[𝑓

(0) + ℎ

(0) 𝑓

1(0) + ℎ

(0) 𝑓

1(0)

+ ℎ (0) 𝑓

1(0)+ ℎ (0) 𝑓

(𝑖V)1

(0)]𝑡2𝛼

Γ (2𝛼 + 1)

+ [ℎ (0) 𝑓

(0) + 2ℎ

(0) 𝑓

(0)

+ ℎ (0) 𝑓(𝑖V)

(0)]𝑡3𝛼

Γ (3𝛼 + 1)} + ⋅ ⋅ ⋅ ,

(17)

ℎ2(𝑡) = 𝑓

1(0) + [𝑓

(0) + ℎ

(0) 𝑓

1(0)

+ ℎ (0) 𝑓

1(0)]

𝑡𝛼

Γ (𝛼 + 1)

+ {ℎ (0) [𝑓

(0) + ℎ

(0) 𝑓

1(0) + ℎ

(0) 𝑓

1(0)

+ ℎ (0) 𝑓

1(0) + ℎ (0) 𝑓

(𝑖V)1

(0)]

+ ℎ (0) [𝑓

(0) + ℎ

(0) 𝑓

1(0)

+ 2ℎ (0) 𝑓

1(0)

+ ℎ (0) 𝑓

1(0) + ℎ

(0) 𝑓

1(0)

+ 2ℎ (0) 𝑓(𝑖V)1

(0)

+ ℎ (0) 𝑓(V)1

(0)]}𝑡2𝛼

Γ (2𝛼 + 1)

+ {ℎ (0) [ℎ

(0) 𝑓

(0) + 2ℎ

(0) 𝑓

(0)

+ ℎ (0) 𝑓(𝑖V)

(0) ]

+ ℎ (0) [ℎ

(0) 𝑓

(0) + 3ℎ

(0) 𝑓

(0)

+ 3ℎ (0) 𝑓(𝑖V)

(0)

+ ℎ (0) 𝑓(V)

(0) ]}𝑡3𝛼

Γ (3𝛼 + 1)+ ⋅ ⋅ ⋅ .

(18)

Equating (15) and (17) yields the following:

ℎ1(0) = 𝑓

1(0) ,

ℎ1(0) = 𝑓 (0) + ℎ (0) 𝑓

1(0) ,

ℎ1(0) = ℎ (0) [𝑓

(0) + ℎ

(0) 𝑓

1(0) + ℎ

(0) 𝑓

1(0)

+ ℎ (0) 𝑓

1(0) + ℎ (0) 𝑓

(𝑖V)1

(0)] ,

...

(19)

and equating (16) and (18) yields the following:

ℎ2(0) = 𝑓

1(0) ,

ℎ2(0) = 𝑓

(0) + ℎ

(0) 𝑓

1(0) + ℎ (0) 𝑓

1(0) ,

(20)

ℎ2(0) = ℎ

(0) [𝑓

(0) + ℎ

(0) 𝑓

1(0) + ℎ

(0) 𝑓

1(0)

+ℎ (0) 𝑓

1(0) + ℎ (0) 𝑓

(𝑖V)1

(0)]

+ ℎ (0) [𝑓

(0) + ℎ

(0) 𝑓

1(0) + 2ℎ

(0) 𝑓

1(0)

+ ℎ (0) 𝑓

1(0) + ℎ

(0) 𝑓

1(0)

+2ℎ (0) 𝑓(𝑖V)1

(0) + ℎ (0) 𝑓(V)1

(0)] ,

...(21)

Using the above data in the following Taylor series expansionof unknown function 𝑓(𝑥) we get

𝑓 (𝑥) = 𝑓 (0) + 𝑓

(0) 𝑥 + 𝑓

(0)𝑥2

2!+ 𝑓 (0)

𝑥3

3!+ ⋅ ⋅ ⋅ .

(22)

Consequently, we determine 𝑓(𝑥) as follows:

𝑓 (𝑥) = [ℎ

1(0) − ℎ (0) 𝑓

1(0)]

+ [ℎ2(0) − ℎ

(0) 𝑓

1(0) − ℎ (0) 𝑓

1(0)] 𝑥


+ [ℎ1(0)

ℎ (0)− ℎ (0) 𝑓

1(0) − ℎ

(0) 𝑓

1(0)

− ℎ (0) 𝑓

1(0) − ℎ (0) 𝑓

(𝑖V)1

(0)]𝑥2

2!+ ⋅ ⋅ ⋅ ,

(23)

where ℎ(0) ̸= 0.

2.2. Determination of Unknown Source Functions Dependingon 𝑡. Weconsider the following inverse problemof determin-ing the source function 𝑓(𝑡):

𝐷𝛼𝑡𝑢 (𝑥, 𝑡) = ℎ (𝑥) 𝑢

𝑥𝑥(𝑥, 𝑡) + 𝑓 (𝑡) ,

𝑥 > 0, 𝑡 > 0, 0 < 𝛼 ≤ 1,

𝑢 (𝑥, 0) = 𝑓1(𝑥) ,

𝑢 (0, 𝑡) = ℎ1(𝑡) ,

𝑢𝑥(0, 𝑡) = ℎ

2(𝑡) ,

(24)

where ℎ1(𝑡), ℎ2(𝑡) ∈ 𝐶∞[0,∞), ℎ(𝑥), 𝑓

1(𝑥) ∈ 𝐶∞[0,∞), and

𝑓(𝑡) ∈ 𝐶1𝜇[0,∞), 𝜇 ≥ −1. As in the previous case, we apply

ADM to determine the unknown function 𝑓(𝑡).First, to reduce the problem, we define new functions in

the following form:

𝑤 (𝑡) = 𝐽𝛼

𝑡𝑓 (𝑡)

𝑢 (𝑥, 𝑡) = V (𝑥, 𝑡) + 𝑤 (𝑡) .(25)

Then the reduced problem is given as follows:

𝐷𝛼𝑡V (𝑥, 𝑡) = ℎ (𝑥) V

𝑥𝑥(𝑥, 𝑡) , (26)

with the following initial and mixed boundary conditions

V (𝑥, 0) = 𝑓1(𝑥) − 𝑤 (0) ,

V (0, 𝑡) = ℎ1(𝑡) − 𝑤 (𝑡) ,

V𝑥(0, 𝑡) = ℎ

2(𝑡) .

(27)

By using ADM as in the previous section, we determine thefunction 𝑤(𝑡) which leads to the source function 𝑓(𝑡). Let usapply 𝐽𝛼

𝑡to both sides of (26) as shown below

𝐽𝛼𝑡𝐷𝛼𝑡V (𝑥, 𝑡) = 𝐽𝛼

𝑡(ℎ (𝑥) V

𝑥𝑥(𝑥, 𝑡)) . (28)

Then we get

V (𝑥, 𝑡) = V (𝑥, 0) + 𝐽𝛼𝑡(ℎ (𝑥) V

𝑥𝑥(𝑥, 𝑡)) . (29)

Now we define the solution V(𝑥, 𝑡) by the following decom-position series according to ADM

V (𝑥, 𝑡) =∞

∑𝑛=0

V𝑛(𝑥, 𝑡) . (30)

Substituting (30) into (29), we obtain

V0(𝑥, 𝑡) = V (𝑥, 0) ,

V𝑛+1

(𝑥, 𝑡) = 𝐽𝛼

𝑡(ℎ (𝑥) (V

𝑛)𝑥𝑥

(𝑥, 𝑡)) , 𝑛 = 0, 1, . . . .(31)

Hence, the recurrence scheme is obtained as follows:V0(𝑥, 𝑡) = V (𝑥, 0) = 𝑓

1(𝑥) − 𝑤 (0) ,

V1(𝑥, 𝑡) = 𝐽

𝛼

𝑡(ℎ (𝑥) (V

0)𝑥𝑥

(𝑥)) = ℎ (𝑥) 𝑓

1(𝑥)

𝑡𝛼

Γ (𝛼 + 1),

V2(𝑥, 𝑡) = 𝐽

𝛼

𝑡(ℎ (𝑥) (V

1)𝑥𝑥

(𝑥)) = ℎ2

(𝑥) 𝑓(𝑖V)1

(𝑥)𝑡2𝛼

Γ (2𝛼 + 1),

...(32)

Consequently, from (30), the solution V(𝑥, 𝑡) is given as shownbelow

V (𝑥, 𝑡) = V0(𝑥, 𝑡) + V

1(𝑥, 𝑡) + V

2(𝑥, 𝑡) + ⋅ ⋅ ⋅

= 𝑓1(𝑥) − 𝑤 (0) + ℎ (𝑥) 𝑓

1(𝑥)

𝑡𝛼

Γ (𝛼 + 1)

+ ℎ2 (𝑥) 𝑓(𝑖V)1

(𝑥)𝑡2𝛼

Γ (2𝛼 + 1)+ ⋅ ⋅ ⋅ .

(33)

By using the boundary condition V(0, 𝑡) = ℎ1(𝑡) + 𝑤(𝑡) and

𝑤(0) = 0, we have

𝑤 (𝑡) = 𝑓1(0) − ℎ

1(𝑡) + ℎ (0) 𝑓

1(0)

𝑡𝛼

Γ (𝛼 + 1)

+ ℎ2 (0) 𝑓(𝑖V)1

(0)𝑡2𝛼

Γ (2𝛼 + 1)+ ⋅ ⋅ ⋅ ,

(34)

which implies the following:

𝐽𝛼𝑡𝑓 (𝑡) = 𝑓

1(0) − ℎ

1(𝑡) + ℎ (0) 𝑓

1(0)

𝑡𝛼

Γ (𝛼 + 1)

+ ℎ2 (0) 𝑓(𝑖V)1

(0)𝑡2𝛼

Γ (2𝛼 + 1)+ ⋅ ⋅ ⋅ .

(35)

Since 𝐷𝛼𝑡𝑤(𝑥, 𝑡) = 𝐷𝛼

𝑡𝐽𝛼𝑡𝑓(𝑡) = 𝑓(𝑡), we obtain the source

function 𝑓(𝑡) as follows:

𝑓 (𝑡) = 𝐷𝛼

𝑡[𝑓1(𝑥) − ℎ

1(𝑡) + ℎ (𝑥) 𝑓

1(𝑥)

𝑡𝛼

Γ (𝛼 + 1)

+ ℎ2 (𝑥) 𝑓(𝑖V)1

(𝑥)𝑡2𝛼

Γ (2𝛼 + 1)+ ⋅ ⋅ ⋅ ] .

(36)

3. Examples

Example 1. We consider the inverse problem of determiningsource function 𝑓(𝑥) in the following one-dimensional frac-tional heat-like PDE:

𝐷𝛼𝑡𝑢 (𝑥, 𝑡) = 2𝑢

𝑥𝑥(𝑥, 𝑡) + 𝑓 (𝑥) , 𝑥 > 0, 0 < 𝛼 ≤ 1, 𝑡 > 0,

(37)


subject to the following initial and nonhomogeneous mixedboundary conditions:

𝑢 (𝑥, 0) = 𝑒𝑥 + sin𝑥,

𝑢 (0, 𝑡) = 𝑒2𝑡,

𝑢𝑥(0, 𝑡) = 𝑒

2𝑡+1.

(38)

Now, let us apply the time-dependent Riemann Liouvillefractional integral operator 𝐽𝛼

𝑡to both sides of (37)

𝐽𝛼𝑡𝐷𝛼𝑡𝑢 (𝑥, 𝑡) = 2𝐽

𝛼

𝑡𝑢𝑥𝑥

(𝑥, 𝑡) + 𝐽𝛼

𝑡𝑓 (𝑥) (39)

which implies

𝑢 (𝑥, 𝑡) − 𝑢 (𝑥, 0) = 2𝐽𝛼

𝑡𝑢𝑥𝑥

(𝑥, 𝑡) + 𝑓 (𝑥) 𝐽𝛼

𝑡(1) . (40)

Then, from the initial condition we get

𝑢 (𝑥, 𝑡) = 𝑒𝑥 + sin𝑥 + 𝑓 (𝑥) 𝑡

𝛼

Γ (𝛼 + 1)+ 2𝐽𝛼𝑡𝑢𝑥𝑥

(𝑥, 𝑡) . (41)

Now, we apply ADM to the problem. In (41), the sum of thefirst three terms is identified as 𝑢

0. So

𝑢0= 𝑒𝑥 + sin𝑥 + 𝑓 (𝑥) 𝑡

𝛼

Γ (𝛼 + 1),

𝑢𝑘+1

= 2𝐽𝛼𝑡(𝑢𝑘)𝑥𝑥

(𝑥, 𝑡) , 𝑘 ≥ 0.

(42)

For 𝑘 = 0, we have

𝑢1= 2𝐽𝛼𝑡(𝑢0)𝑥𝑥

(𝑥, 𝑡)

= 2𝑒𝑥𝑡𝛼

Γ (𝛼 + 1)− 2 sin𝑥 𝑡

𝛼

Γ (𝛼 + 1)

+ 2𝑓 (𝑥)𝑡2𝛼

Γ (2𝛼 + 1)+ ⋅ ⋅ ⋅ ,

(43)

similarly, for 𝑘 = 1, we have

𝑢2= 2𝐽𝛼𝑡(𝑢1)𝑥𝑥

(𝑥, 𝑡)

= 4𝑒𝑥𝑡2𝛼

Γ (2𝛼 + 1)+ 4 sin𝑥 𝑡

2𝛼

Γ (2𝛼 + 1)

+ 4𝑓(𝑖V) (𝑥)𝑡3𝛼

Γ (3𝛼 + 1)+ ⋅ ⋅ ⋅ ,

(44)

and for 𝑘 = 2, we have

𝑢3=2𝐽𝛼𝑡(𝑢2)𝑥𝑥

(𝑥, 𝑡)

= 8𝑒𝑥𝑡3𝛼

Γ (3𝛼 + 1)− 8 sin𝑥 𝑡

3𝛼

Γ (3𝛼 + 1)

+ 8𝑓(V𝑖) (𝑥)𝑡4𝛼

Γ (4𝛼 + 1)+ ⋅ ⋅ ⋅

...

(45)

Then using ADM polynomials, we get the solution 𝑢(𝑥, 𝑡) asfollows:𝑢 (𝑥, 𝑡) = 𝑢

0+ 𝑢1+ 𝑢2+ 𝑢3+ ⋅ ⋅ ⋅

= 𝑒𝑥 + sin𝑥 + 𝑓 (𝑥) 𝑡𝛼

Γ (𝛼 + 1)+ 2𝑒𝑥

𝑡𝛼

Γ (𝛼 + 1)

− 2 sin𝑥 𝑡𝛼

Γ (𝛼 + 1)+ 2𝑓 (𝑥)

𝑡2𝛼

Γ (2𝛼 + 1)

+ 4𝑒𝑥𝑡2𝛼

Γ (2𝛼 + 1)+ 4 sin𝑥 𝑡

2𝛼

Γ (2𝛼 + 1)

+ 4𝑓(𝑖V) (𝑥)𝑡3𝛼

Γ (3𝛼 + 1)+ 8𝑒𝑥

𝑡3𝛼

Γ (3𝛼 + 1)

− 8 sin𝑥 𝑡3𝛼

Γ (3𝛼 + 1)+ 8𝑓(V𝑖) (𝑥)

𝑡4𝛼

Γ (4𝛼 + 1)+ ⋅ ⋅ ⋅ .

(46)

After arranging it according to like powers of 𝑡, we have

𝑢 (𝑥, 𝑡) = 𝑒𝑥 + sin𝑥 + 𝑡

𝛼

Γ (𝛼 + 1)[𝑓 (𝑥) + 2𝑒

𝑥 − 2 sin𝑥]

+𝑡2𝛼

Γ (2𝛼 + 1)[2𝑓 (𝑥) + 4𝑒

𝑥 + 4 sin𝑥]

+𝑡3𝛼

Γ (3𝛼 + 1)[4𝑓(𝑖V) (𝑥) + 8𝑒

𝑥 − 8 sin𝑥]

+𝑡4𝛼

Γ (4𝛼 + 1)[8𝑓(V𝑖) (𝑥) + 16𝑒

𝑥 + 16 sin𝑥] + ⋅ ⋅ ⋅ .

(47)

Now, by applying the boundary condition given in (38), weobtain

𝑢 (0, 𝑡) = 1 +𝑡𝛼

Γ (𝛼 + 1)[𝑓 (0) + 2] +

𝑡2𝛼

Γ (2𝛼 + 1)[2𝑓 (0) + 4]

+𝑡3𝛼

Γ (3𝛼 + 1)[4𝑓(𝑖V) (0) + 8]

+𝑡4𝛼

Γ (4𝛼 + 1)[8𝑓(V𝑖) (0) + 16] + ⋅ ⋅ ⋅ .

(48)

From (15), it must be equal to the following Taylorseries expansion of 𝑒2𝑡 in the space whose bases are{𝑡𝑛𝛼/Γ(𝑛𝛼 + 1)}

∞

𝑛=0, 0 < 𝛼 ≤ 1:

𝑒2𝑡 = 1 + 2𝑡𝛼

Γ (𝛼 + 1)+ 4

𝑡2𝛼

Γ (2𝛼 + 1)

+ 8𝑡3𝛼

Γ (3𝛼 + 1)+ 16

𝑡4𝛼

Γ (4𝛼 + 1)+ ⋅ ⋅ ⋅ .

(49)

Hence, from the equality of the coefficients of correspondingterms, we get

𝑓 (0) = 𝑓

(0) = 𝑓(𝑖V)

(0) = 𝑓(V𝑖)

(0) = ⋅ ⋅ ⋅ = 0. (50)


From (47), we have

𝑢𝑥(𝑥, 𝑡) = 𝑒

𝑥 + cos𝑥 + 𝑡𝛼

Γ (𝛼 + 1)[𝑓 (𝑥) + 2𝑒

𝑥 − 2 cos𝑥]

+𝑡2𝛼

Γ (2𝛼 + 1)[2𝑓 (𝑥) + 4𝑒

𝑥 + 4 cos𝑥]

+𝑡3𝛼

Γ (3𝛼 + 1)[4𝑓(V) (𝑥) + 8𝑒

𝑥 − 8 cos𝑥]

+𝑡4𝛼

Γ (4𝛼 + 1)[8𝑓(V𝑖𝑖) (𝑥) + 16𝑒

𝑥 + 16 cos𝑥] + ⋅ ⋅ ⋅ .

(51)

So,

𝑢𝑥(0, 𝑡) = 2 +

𝑡𝛼

Γ (𝛼 + 1)𝑓 (0) +

𝑡2𝛼

Γ (2𝛼 + 1)[2𝑓 (0) + 8]

+𝑡3𝛼

Γ (3𝛼 + 1)[4𝑓(V) (0)]

+𝑡4𝛼

Γ (4𝛼 + 1)[8𝑓(V𝑖𝑖) (0) + 32] + ⋅ ⋅ ⋅ .

(52)

From the derivative boundary condition given in (38), it mustbe equal to the following series expansion of 𝑒2𝑡+1 in the spacewhose bases are {𝑡𝑛𝛼/Γ(𝑛𝛼 + 1)}∞

𝑛=0, 0 < 𝛼 ≤ 1:

𝑒2𝑡 + 1 = 2 + 2𝑡𝛼

Γ (𝛼 + 1)+ 4

𝑡2𝛼

Γ (2𝛼 + 1)

+ 8𝑡3𝛼

Γ (3𝛼 + 1)+ 16

𝑡4𝛼

Γ (4𝛼 + 1)+ ⋅ ⋅ ⋅ .

(53)

Then, we find the following data:

𝑓 (0) = 2, 𝑓

(0) = −2, 𝑓(V)

(0) = 2,

𝑓(V𝑖𝑖) (0) = −2, . . . .(54)

Next, using (50) and (54), we have the Taylor series expansionof 𝑓(𝑥) as follows:

𝑓 (𝑥) = 𝑓 (0) + 𝑓

(0) 𝑥 + 𝑓

(0)𝑥2

2!

+ 𝑓 (0)𝑥3

3!+ 𝑓(𝑖V) (0)

𝑥4

4!+ ⋅ ⋅ ⋅

= 2𝑥 − 2𝑥3

3!+ 2

𝑥5

5!+ 2

𝑥7

7!+ ⋅ ⋅ ⋅ .

(55)

That is,

𝑓 (𝑥) = 2 [𝑥 −𝑥3

3!+𝑥5

5!+𝑥7

7!+ ⋅ ⋅ ⋅ ] (56)

which is the series expansion of the function 2 sin𝑥. Conse-quently, we determine the source function 𝑓(𝑥) as

𝑓 (𝑥) = 2 sin𝑥. (57)

Example 2. We consider the inverse problem of determiningsource function 𝑓(𝑡) in the following one-dimensional frac-tional heat-like diffusion equation:

𝐷𝛼𝑡𝑢 (𝑥, 𝑡) =

1

2𝑥2𝑢𝑥𝑥

(𝑥, 𝑡) + 𝑓 (𝑡) ,

𝑥 > 0, 0 < 𝛼 ≤ 1, 𝑡 > 0,

(58)

subject to following initial and mixed boundary conditions

𝑢 (𝑥, 0) = 𝑥2 +

1

2, 𝑢 (0, 𝑡) =

𝑒2𝑡

2, 𝑢

𝑥(0, 𝑡) = 0.

(59)

Now let us determine the source function 𝑓(𝑡). To reduce theproblem, we define new functions as follows:

𝑤 (𝑡) = 𝐽𝛼

𝑡𝑓 (𝑡) ,

𝑢 (𝑥, 𝑡) = V (𝑥, 𝑡) + 𝑤 (𝑡) .(60)

Then, our reduced problem is given as follows:

𝐷𝛼𝑡V (𝑥, 𝑡) =

1

2𝑥2V𝑥𝑥

(𝑥, 𝑡) , 0 < 𝛼 ≤ 1, 𝑡 > 0

V (𝑥, 0) = 𝑢 (𝑥, 0) − 𝑤 (0) = 𝑥2 +1

2,

V (0, 𝑡) = 𝑢 (0, 𝑡) − 𝑤 (𝑡) =𝑒2𝑡

2− 𝑤 (𝑡) ,

V𝑥(0, 𝑡) = 0.

(61)

Applying 𝐽𝛼𝑡to both sides of (61), then we get

V (𝑥, 𝑡) − V (𝑥, 0) =1

2𝑥2𝐽𝛼𝑡V𝑥𝑥

(𝑥, 𝑡) (62)

which implies

V (𝑥, 𝑡) = 𝑥2 +1

2+1

2𝑥2𝐽𝛼𝑡V𝑥𝑥

(𝑥, 𝑡) . (63)

By using ADM for (63), we obtain

V0= 𝑥2 +

1

2,

V𝑘+1

=1

2𝑥2𝐽𝛼𝑡(V𝑘)𝑥𝑥

(𝑥, 𝑡) , 𝑘 ≥ 0.

(64)

Then, for 𝑘 = 0, we get

V1=

1

2𝑥2𝐽𝛼𝑡(V0)𝑥𝑥

(𝑥, 𝑡)

= 𝑥2𝑡𝛼

Γ (𝛼 + 1),

(65)

similarly, for 𝑘 = 1, we get

V2=

1


(𝑥, 𝑡)

= 𝑥2𝑡2𝛼

Γ (2𝛼 + 1),

(66)


and for 𝑘 = 2, we get

V3=

1


(𝑥, 𝑡)

= 𝑥2𝑡3𝛼

Γ (3𝛼 + 1),

...

(67)

As a result, we get the solution V as follows:

V (𝑥, 𝑡) = V0+ V1+ V2+ V3+ ⋅ ⋅ ⋅

= 𝑥2 +1

2+ 𝑥2

𝑡𝛼

Γ (𝛼 + 1)

+ 𝑥2𝑡2𝛼

Γ (2𝛼 + 1)+ 𝑥2

𝑡3𝛼

Γ (3𝛼 + 1)+ ⋅ ⋅ ⋅

=1

2+ 𝑥2 [1 +

𝑡𝛼

Γ (𝛼 + 1)+

𝑡2𝛼

Γ (2𝛼 + 1)

+𝑡3𝛼

Γ (3𝛼 + 1)+ ⋅ ⋅ ⋅ ] .

(68)

Therefore, from the boundary condition we have

𝑤 (𝑡) =𝑒2𝑡

2−1

2. (69)

Using (69) in the definition 𝐷𝛼𝑡𝑤(𝑡) = 𝐷𝛼

𝑡𝐽𝛼𝑡𝑓(𝑡) =

𝑓(𝑡), finally we obtain the source function 𝑓(𝑡) as 𝑓(𝑡) =𝐷𝛼𝑡((𝑒2𝑡/2) − (1/2)); that is,

𝑓 (𝑡) =1

2𝐷𝛼𝑡(𝑒2𝑡) . (70)

Here,

𝐷𝛼𝑡(𝑒2𝑡) = 𝑡−𝛼𝐸

1,1−𝛼(2𝑡) , (71)

where 𝐸1,1−𝛼

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

[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theoryand Applications of Fractional Differential Equations, vol. 204of North-Holland Mathematics Studies, Elsevier Science B.V.,Amsterdam, The Netherlands, 2006.

[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.

[4] I. Podlubny, Fractional Differential Equations, vol. 198 ofMath-ematics in Science and Engineering, Academic Press, San Diego,Calif, USA, 1999.

[5] S. S. Ray, K. S. Chaudhuri, and R. K. Bera, “Analytical approx-imate solution of nonlinear dynamic system containing frac-tional derivative by modified decomposition method,” AppliedMathematics and Computation, vol. 182, no. 1, pp. 544–552,2006.

[6] K. Diethelm, “An algorithm for the numerical solution of differ-ential equations of fractional order,” Electronic Transactions onNumerical Analysis, vol. 5, no. Mar., pp. 1–6, 1997.

[7] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, FractionalCalculus Models and Numerical Methods, vol. 3 of Series onComplexity, Nonlinearity and Chaos, World Scientific Publish-ing, Hackensack, NJ, USA, 2012.

[8] M. Alipour and D. Baleanu, “Approximate analytical solutionfor nonlinear system of fractional differential equations by BPsoperational matrices,” Advances in Mathematical Physics, vol.2013, Article ID 954015, 9 pages, 2013.

[9] G. Adomian, Solving Frontier Problems of Physics: The Decom-position Method, vol. 60 of Fundamental Theories of Physics,Kluwer Academic Publishers, Dordrecht, The Netherlands,1994.

[10] G. Adomian, “Solutions of nonlinear P. D. E,” Applied Mathe-matics Letters, vol. 11, no. 3, pp. 121–123, 1998.

[11] K. Abbaoui and Y. Cherruault, “The decomposition methodapplied to the Cauchy problem,” Kybernetes, vol. 28, no. 1, pp.68–74, 1999.

[12] D. Kaya and A. Yokus, “A numerical comparison of partialsolutions in the decompositionmethod for linear and nonlinearpartial differential equations,” Mathematics and Computers inSimulation, vol. 60, no. 6, pp. 507–512, 2002.

[13] D. A. Murio, “Time fractional IHCP with Caputo fractionalderivatives,” Computers & Mathematics with Applications, vol.56, no. 9, pp. 2371–2381, 2008.

[14] A. N. Bondarenko and D. S. Ivaschenko, “Numerical methodsfor solving inverse problems for time fractional diffusion equa-tion with variable coefficient,” Journal of Inverse and Ill-PosedProblems, vol. 17, no. 5, pp. 419–440, 2009.

[15] Y. Zhang and X. Xu, “Inverse source problem for a fractionaldiffusion equation,” Inverse Problems, vol. 27, no. 3, Article ID035010, 12 pages, 2011.

[16] M. Kirane and S. A. Malik, “Determination of an unknownsource term and the temperature distribution for the linear heatequation involving fractional derivative in time,”Applied Math-ematics and Computation, vol. 218, no. 1, pp. 163–170, 2011.

[17] M. Caputo, “Linear models of dissipation whose Q is almostfrequency independent—II,” Geophysical Journal International,vol. 13, no. 5, pp. 529–539, 1967.

[18] S. T. Mohyud-Din, A. Yıldırım, and M. Usman, “Homotopyanalysis method for fractional partial differential equations,”International Journal of Physical Sciences, vol. 6, no. 1, pp. 136–145, 2011.

[19] G. Mittag-Leffler, “Sur la représentation analytique d’unebranche uniforme d’une fonction monogène,” Acta Mathemat-ica, vol. 29, no. 1, pp. 101–181, 1905.

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