Analytical Solution of the Fractional Fredholm Integrodifferential Equation Using the Fractional Residual Power Series Method

Abstract

We study the solution of fractional Fredholm integrodifferential equation. A modified version of the fractional power series method (RPS) is presented to extract an approximate solution of the model. The RPS method is a combination of the generalized fractional Taylor series and the residual functions. To show the efficiency of the proposed method, numerical results are presented.

1. Introduction

Fractional Fredholm integrodifferential equations have several applications in sciences and engineering. The closed form of the exact solution of such problems is difficult to find and in most of the cases is not available. For this reason, researchers are looking for the numerical solutions of such problems. Irandoust-Pakchin and Abdi-mazraeh [1] used the variational iteration method for solving fractional integrodifferential equations with the nonlocal boundary conditions. Adomian decomposition method is used in [2, 3] while the homotopy perturbation method is used in [4, 5]. Wazwaz [6–8] studied the Fredholm integral equations of the formwhere and are constants, is a parameter, is a smooth function as the discussion required, and is the kernel. In this paper, we study the generalization of the above problem. We study the following class of fractional Fredholm integrodifferential equations of the formsubject toThe fractional derivative in (2) is in the Caputo sense. If , we do not need the initial condition (3) and we return back to the problem which is discussed by Wazwaz [8]. In the following definition and theorem, we write the definition of Caputo derivative as well as the power rule which we are using in this paper. For more details on the geometric and physical interpretation for Caputo fractional derivatives, see [9].

Definition 1. For to be the smallest integer that exceeds , the Caputo fractional derivatives of order are defined as

Theorem 2. The Caputo fractional derivative of the power function satisfies

Definition 3. The Riemann-Liouville fractional integral operator of order α of , is defined aswhere is the space of real functions , such that, for each , there exists a real number such that where
In addition, if where . We present the following definition and some properties of the fractional power series which are used in this paper. More details can be found in [10].

Definition 4. A power series expansion of the formwhere , is called fractional power series FPS about. .

Theorem 5. Suppose that has a fractional FPS representation at of the formIf , are continuous on , then .

Theorem 6. Let and for where . Then,

Theorem 7. Let , , and can differentiate with respect to for where Then,

Theorem 8. Let on where . Then, the reminder satisfies

This paper is organized as follows. A description of the modified fractional power series method (MFPS) for approximating the fractional Fredholm integrodifferential equations problem (2)-(3) is presented in Section 2. Several numerical examples are discussed in Section 3. Conclusions and closing remarks are given in Section 4.

2. Algorithm of the MFPS Method

Consider the following class of fractional Fredholm integrodifferential equations of the formsubject toUsing the MFPS method, the solution problem (2)-(3) can be written the fractional power series form asTo obtain the approximate values of the above series (14), the th truncated series is written in the formSince , we rewrite (14) aswhere is considered to be the 0th MRPS approximate solution of To find the values of the MFPS-coefficients , we solve the fractional differential equationwhere is the th residual function and is defined byTo determine the coefficient in the expansion (15), we substitute the 1st RPS approximate solutioninto (18) to getThen, we solve to getTo find , we substitute the 2nd RPS approximate solutioninto the 2nd residual function such thatThen, we solve to getTo find , we substitute the 3rd RPS approximate solution into the 3rd residual function such thatThen, we solve to getTo find , we substitute the 4th RPS approximate solutioninto the 4th residual function such thatThen, we solve to getUsing similar argument, we find thatThus,For .

3. Numerical Results

In this section, we present three examples to show the efficiency of the proposed method. We use Mathematica software to generate the results in this section.

Example 1. Consider the following fractional Fredholm integrodifferential equation:subject toThe exact solution isUsing the same argument described in the previous section, we find thatThus,which is the exact solution.

Example 2. Consider the following fractional Fredholm integrodifferential equation:subject toThe exact solution isUsing the same argument described in the previous section, we find the first few terms which areContinuing in this process, we find thatThus, if for some positive integer ,Hence,which is the exact solution.

Example 3. Consider the following fractional Fredholm integrodifferential equation:subject toThe exact solution isUsing the same argument described in the previous section, we find that the first few terms areContinuing in this process, we find thatThus,which is the exact solution.

4. Conclusions and Closing Remarks

In this paper we employed the MFPS method to handle the fractional Fredholm integrodifferential problems. The method showed reliability in handling these ill-posed problems. It is worth mentioning that we get the exact solution in the above three examples. We test the prosed method and we get very accurate results. Although the MRPS method is not commonly used for such problems, it gives us very accurate results. This technique can be extended to other applications in science and engineering.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author also would like to express his sincere appreciation to the United Arab Emirates University Research Affairs for the financial support of Grant no. SURE Plus 21.