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On the Inverse Problem of the Fractional Heat-Like Partial Differential Equations: Determination of the Source Function
The study in this paper mainly concerns the inverse problem of determining an unknown source function in the linear fractional differential equation with variable coefficient using Adomian decomposition method (ADM). We apply ADM to determine the continuous right hand side functions and in the heat-like diffusion equations and , respectively. The results reveal that ADM is very effective and simple for the inverse problem of determining the source function.
Fractional differential equations (FDEs) are obtained by generalizing differential equations to an arbitrary order. They are used to model physical systems with memory. Since FDEs have memory, nonlocal relations in space and time complex phenomena can be modeled by using these equations. Due to 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 applications of FDEs. Since FDEs are used to model complex phenomena, they play a crucial role in engineering, physics, and applied mathematics. Therefore, they are generating an increasing interest from engineers and scientist in the recent years. As a result, FDEs are quite frequently encountered in different research areas and engineering applications .
The book written by Oldham and Spanier  played an outstanding role in the development of the fractional calculus. Also, it was the first book that was entirely devoted to a systematic presentation of the ideas, methods, and applications of the fractional calculus. Afterwards, several fundamental works on various aspects of the fractional calculus include extensive survey on fractional differential equations by Miller and Ross , Podlubny , and others. Further, several references to the books by Oldham and Spanier , Miller and Ross , and Podlubny  show that applied scientists need first of all an easy introduction to the theory of fractional derivatives and fractional differential equations, which could help them in their initial steps in adopting the fractional calculus as a method of research .
In general, FDEs do not have exact analytical solutions; hence, the approximate and numerical solutions of these equations are studied [6–8]. Analytical approximations for linear and nonlinear FDEs are obtained by variational iteration method, Adomian decomposition method, homotopy perturbation method, Lagrange multiplier method, BPs operational matrices method, and so forth. An effective and easy-to-use method for solving such equations is needed. Large classes of linear and nonlinear differential equations, both ordinary and partial, can be solved by the Adomian decomposition method [9–12].
Solving an equation with certain data in a specified region is called direct problem. On the other hand, determining an unknown input by using output is called an inverse problem. This unknown input could be some coefficients, or it could be a source function in equation. Based on this unknown input the inverse problem is called inverse problem of coefficient identification or inverse problem of source identification, respectively. Generally inverse problems are ill-posed problems; that is, they are very sensitive to errors in measured input. In order to deal with this ill-posedness, regularization methods have been developed. Inverse problems have many practical applications such as geophysics, optics, quantum mechanics, astronomy, medical imaging and materials testing, X-ray tomography, and photoelasticity. Theoretical and applied aspects of inverse problems have been under intense study lately, especially for the fractional equation [13–16].
In this paper, we investigate inverse problems of the linear heat-like differential equations of fractional orders and where the function is assumed to be a causal function of time and space. Time fractional derivative operator is considered as in Caputo sense . We use the Adomian decomposition method [9, 10] to obtain source functions and under the initial and mixed boundary conditions. By this method, we determine the source functions and in a rapidly converging series form when they exist. Compared with previous researches [10, 12, 18], the method we use in this paper is more effective and accurate.
The structure of this paper is given as follows. First, we give some basic definitions of fractional calculus. Inverse problem of finding the source function in one-dimensional fractional heat-like equations with mixed boundary conditions is given in Section 2. After that, we give some illustrative examples of this method for all cases in Section 3. Finally, the conclusion is given in Section 4.
1.1. Fractional Calculus
Definition 1. A real function , , is said to be in the space , if there exists a real number such that , where , and it is said to be in the space if , .
Definition 2. The Riemann-Liouville fractional integral operator of order , of a function , is defined as Some of the basic properties of this operator are given as follows.
For , , and : The other properties can be found in .
Definition 3. The fractional derivative of in the Caputo sense is defined as where , , , .
Useful properties of are given as follows.
Lemma 4. If , , and , , then Since traditional initial and boundary conditions are allowed in problems including Caputo fractional derivatives, it is considered here. In this paper, we deal with the fractional heat-like equations where the unknown function is an arbitrary function of time and space.
Definition 5. The Caputo time fractional derivative operator of order is defined as follows where is the smallest integer that exceeds : For more details about Caputo fractional differential operator, we refer to .
Definition 6. The Mittag-Leffler function with two-parameters is defined by the series expansion as shown below, where the real part of is strictly positive 
2. Inverse Problem of Determining Source Function
In this section, we deal with inverse problem of finding the source function, in one-dimensional fractional heat-like equations with mixed boundary conditions. To determine the unknown source function we have developed new methods through ADM as in the following subsections.
2.1. Determination of Unknown Source Functions Depending on
We consider the following inverse problem of determining the source function : where the functions , and , , . In order to determine the source function for this kind of inverse problems, we apply ADM. First, we apply the time-dependent Riemann-Liouville fractional integral operator to both sides of (7) to get rid of fractional derivative as shown below: Then we get In ADM the solution is written in the following series form : where and , , are defined in . After substituting the decomposition (10) into (9) and setting the recurrence scheme as follows: we get ADM polynomials below After writing these polynomials in (10), the solution is given by If we arrange it with respect to like powers of , then we get To determine the unknown source function, first we expand the boundary conditions and into the following series for the space whose bases are , : On the other hand, if we rewrite the boundary conditions and from (14), then we have Equating (15) and (17) yields the following: and equating (16) and (18) yields the following: Using the above data in the following Taylor series expansion of unknown function we get Consequently, we determine as follows: where .
2.2. Determination of Unknown Source Functions Depending on
We consider the following inverse problem of determining the source function : where , , , , and , . As in the previous case, we apply ADM to determine the unknown function .
First, to reduce the problem, we define new functions in the following form: Then the reduced problem is given as follows: with the following initial and mixed boundary conditions By using ADM as in the previous section, we determine the function which leads to the source function . Let us apply to both sides of (26) as shown below Then we get Now we define the solution by the following decomposition series according to ADM Substituting (30) into (29), we obtain Hence, the recurrence scheme is obtained as follows: Consequently, from (30), the solution is given as shown below By using the boundary condition and , we have which implies the following: Since , we obtain the source function as follows:
Example 1. We consider the inverse problem of determining source function in the following one-dimensional fractional heat-like PDE: subject to the following initial and nonhomogeneous mixed boundary conditions: Now, let us apply the time-dependent Riemann Liouville fractional integral operator to both sides of (37) which implies Then, from the initial condition we get Now, we apply ADM to the problem. In (41), the sum of the first three terms is identified as . So For , we have similarly, for , we have and for , we have Then using ADM polynomials, we get the solution as follows: After arranging it according to like powers of , we have Now, by applying the boundary condition given in (38), we obtain From (15), it must be equal to the following Taylor series expansion of in the space whose bases are , : Hence, from the equality of the coefficients of corresponding terms, we get From (47), we have So, From the derivative boundary condition given in (38), it must be equal to the following series expansion of in the space whose bases are , : Then, we find the following data: Next, using (50) and (54), we have the Taylor series expansion of as follows: That is, which is the series expansion of the function . Consequently, we determine the source function as
Example 2. We consider the inverse problem of determining source function in the following one-dimensional fractional heat-like diffusion equation: subject to following initial and mixed boundary conditions Now let us determine the source function . To reduce the problem, we define new functions as follows: Then, our reduced problem is given as follows: Applying to both sides of (61), then we get which implies By using ADM for (63), we obtain Then, for , we get similarly, for , we get and for , we get As a result, we get the solution as follows: Therefore, from the boundary condition we have Using (69) in the definition , finally we obtain the source function as ; that is, Here, where is Mittag-Leffler function with two parameters given as; (6).
The best part of this method is that one can easily apply ADM to the fractional partial differential equations like applying ADM to ordinary differential equations.
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