Discrete Dynamics in Nature and Society

Volume 2017, Article ID 5234151, 10 pages

https://doi.org/10.1155/2017/5234151

## An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations

Department of Mathematical Sciences, United Arab Emirates University, Al-Ain, UAE

Correspondence should be addressed to Mohammed Al-Refai; ea.ca.ueau@iaferla_m

Received 24 January 2017; Accepted 21 February 2017; Published 8 March 2017

Academic Editor: Thabet Abdeljawad

Copyright © 2017 Moh’d Khier Al-Srihin and Mohammed Al-Refai. 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.

#### Abstract

In this paper, we introduce an efficient series solution for a class of nonlinear multiterm fractional differential equations of Caputo type. The approach is a generalization to our recent work for single fractional differential equations. We extend the idea of the Taylor series expansion method to multiterm fractional differential equations, where we overcome the difficulty of computing iterated fractional derivatives, which are difficult to be computed in general. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed. Several examples are presented to illustrate the efficiency of the new approach and comparison with the Adomian decomposition method is performed.

#### 1. Introduction

Fractional differential equations (FDEs) are generalization to differential equations (DEs) for noninteger orders. In recent years, FDEs caught the attention of many researchers because of their appearance in modeling several phenomenon in the physical sciences. As many FDEs do not possess exact solutions on closed forms, analytical and numerical techniques have been implemented to study these equations. Iterative methods, such as the variational iteration method (VIM) in [1], the homotopy analysis method (HAM) in [2, 3], the Adomian decomposition method (ADM) in [4–9], and the fractional differential transform method in [10], have been implemented to solve various types of FDEs. These methods produce a solution in a series form whose terms are determined sequentially. We refer the reader to [11, 12] for a comprehensive study of series solutions of fractional differential equations. Recently, we have introduced a new series solution for single fractional differential equations [13]. The new approach is a modified form of the well-known Taylor series expansion, where we overcome the difficulty of computing iterative fractional derivatives. The efficiency of the new approach has been illustrated through several examples. In this paper we extend the idea to multiterm fractional differential equations. The presented work is a part of the Master thesis [14]. We consider the left Caputo fractional derivative , defined by [15, 16]provided the integral exists, where is the well-known Gamma function. The left Riemann-Liouville fractional integral, , of order , is defined byThe left Caputo derivative is related to the left Riemann-Liouville fractional integral byIt is known thatAlso, for , it holds thatThis paper is organized as follows. In Section 2, we present the series solution of nonlinear two-term fractional differential equations. We illustrate the efficiency of the presented technique through several examples. We also compare our results with the ones obtained by the Adomian decomposition method. In Section 3, we present and illustrate the efficiency of the new series solution for three-term fractional differential equations of several types. Finally, we conclude with some remarks in Section 4.

#### 2. Two-Term Fractional Differential Equations

We start with the nonlinear two-term fractional initial value problems of the formwithwhere , and and are nonzero constants. We assume that is continuous and smooth with respect to . We also assume that and are rational numbers with and , .

##### 2.1. The Expansion Procedure

Let ; we have for some , .

In the following we expand the solution of problem (6)-(7) in an infinite series of the form where the coefficients : have to be determined sequentially in the following manner: From the initial condition (7) we have . Since , for being constant, we havewhere

By substituting (9) in (6) we haveApplying the well-known Taylor series method to compute the coefficients will lead to computing iterated fractional derivatives, which are not easily computed in general. To avoid this difficulty, let ; we haveShifting the index to zero yieldsTo avoid the singularity at , we multiply (13) by ; we haveNow, since , thus , and (14) has no singularity at

Let , and ; then (14) can be written asWe first determine the coefficients for . By performing the th derivative of (15) with respect to and substituting , we havewhich yieldsSince , and is smooth, then for , we have and hence , for .

We now determine , for . By performing the th derivative of (15) with respect to and substituting , we haveUsing the well-known Leibniz rule for differentiating the products, we have

Since we haveFrom the last equation we determine and thus the solution is obtained.

*Remark 1. *In (6), assuming , then and Thus (17) is reduced to which coincides with the coefficients obtained by the Taylor series expansion method. Same comment applies for the coefficient in (22).

*Remark 2. *The algorithm can be generalized for the two-term fractional differential equation (6) with , for arbitrary integer . But we have to take care of the existence of the fractional derivative (see Eq. (5)), by choosing the coefficient , for . This case is discussed in Section 3.

##### 2.2. Numerical Results

*Example 3. *Consider the nonlinear two-term fractional initial value problemwithThe exact solution of the problem is .

Applying the current algorithm, we have , , , , and . We expand the solution in an infinite series of the formThe initial condition in (26) yields . We have andSince is continuous and smooth with respect to , we have Thus . The function satisfies the assumption of the proposed algorithm, and it holds thatThe computation above is made using the software Mathematica version 9. For ; substituting (30) in (22) yieldswhereApplying (31) together with , we have Thus,and the exact solution of problem (25)-(26) is obtained.

In the following we compare our results with the Adomian decomposition method (ADM). Assume that the nonlinear function and the solution can be expressed in the following series form: where are the well-known Adomian polynomials that can be derived from [17] For recent advances in the Adomian decomposition method we refer the reader to [18, 19]. For , we haveApplying the Riemann-Liouville fractional integral operator to (25) and substituting we have

Setand balancing (39) yieldsEvaluating the first 5 term of the power series solution, we obtain the approximate solution , where which is not the exact solution. Solving the problem by the ADM with yields , wherewhich is again not the exact solution. Figure 1 depicts the exact solution obtained by the current algorithm and the approximate solutions and obtained by the Adomian decomposition method.