Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 5234151, 10 pages
https://doi.org/10.1155/2017/5234151

Research Article

An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations

Moh’d Khier Al-Srihin and Mohammed Al-Refai

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

Correspondence should be addressed to Mohammed Al-Refai; m_alrefai@uaeu.ac.ae

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 by

The left Caputo derivative is related to the left Riemann-Liouville fractional integral by

It is known that

Also, for , it holds that

This 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 form

with

where , 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 have

where

By substituting (9) in (6) we have

Applying 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 have

Shifting the index to zero yields

To avoid the singularity at , we multiply (13) by ; we have

Now, since , thus , and (14) has no singularity at

Let , and ; then (14) can be written as

We first determine the coefficients for . By performing the th derivative of (15) with respect to and substituting , we have

which yields

Since , 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 have

Using the well-known Leibniz rule for differentiating the products, we have

Since

we have

From 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 problem

with

The exact solution of the problem is .
Applying the current algorithm, we have , , , , and . We expand the solution in an infinite series of the form

The initial condition in (26) yields . We have and

Since is continuous and smooth with respect to , we have

Thus . The function satisfies the assumption of the proposed algorithm, and it holds that

The computation above is made using the software Mathematica version 9. For ; substituting (30) in (22) yields

where

Applying (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 have

Applying the Riemann-Liouville fractional integral operator to (25) and substituting

we have

Set

and balancing (39) yields

Evaluating 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 , where

which 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.

Figure 1: Comparison between the exact and approximate solutions obtained by the ADM for problem ((25)-(26)).

Example 4. Consider the nonlinear two-term fractional initial value problem

with

This example has been discussed in [20], where the problem is transformed to a fractional integral equation, and then the Adams-Bashforth-Moulton method is used with step size to approximate the solution.
Applying the current algorithm, we have , and

where

The initial condition yields . We apply (46) to compute the first 13 nonzero terms of , and we have

Since the exact solution of problem (44)-(45) is not available in a closed form, we define the error by

where

Tables 1 and 2 present the error for different values of . One can see that the error decreases with and more accuracy can be achieved by considering more terms. Also, the error increases with , as any other series solutions.

Table 1: The error of Example 4 for and different values of .

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Table 2: The error of Example 4 at and for different values of

5

10

15

20

25

30

35
0.0000967893

3. Three-Term Fractional Differential Equations

We consider the nonlinear three-term fractional initial value problem of the form

with

where

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 , where . Let ; we have for , and .

Applying analogous steps for the case of the two-term fractional equations, we seek a solution of the form

From the initial condition (52) we have , and thus, .

Using the fact that , for , we have

where , , and

Substituting (55) in (51), we have

Since the coefficients , are known, we write the last equation in the form

Let ; then (57) reduces to

Substituting and

and shifting the indices to zero, we have

To avoid singularity we multiply both sides in (60) by ; we have

Equation (61) can be written as

where and

Applying analogs steps for deriving (22), we have

where From the last recursion we determine

Remark 5. Our derivation is based on the facts that the nonlinear function is smooth and the fractional differential equations is of constant coefficients. In case if one of these conditions does not hold, a modified treatment will be considered as we will see in Example 7.

3.1. Numerical Results

Example 6. Consider the Bagely-Torvik initial value problem

where

This example has been discussed in [21] using a Chebyshev spectral method, where the solution has been approximated by the shifted Chebyshev polynomials with different degrees. Then the exact solution was obtained by considering the shifted Chebyshev polynomial of degree 9. We mention here that there are simple typos in presenting the example in (64) and in (66) and we correct them here. Applying the current algorithm we have

We expand the solution in infinite series of the form . The initial condition in (65) yields . Let ; applying (59) we have

and it holds that

Using (63) and (69), we have

where

Since is smooth, then , for We now apply the last recursion together with , to compute , for . For , we have

and thus .
Applying analogous arguments yields . For , we have

which, together with , implies that . For , we have

but and ; thus . Applying the same steps yields . For , we have

and thus .
Proceeding in the same manner, we have for . Thus,

and the exact solution of problem (64)-(65) is obtained.

In the following example we show that the current algorithm can be applied to more general multiterm fractional differential equations which are not necessary of the form in (51).

Example 7. Consider the nonlinear three-term fractional initial value problem

with

The exact solution for this problem is .

Applying the current algorithm we have . We then expand the solution in an infinite series of the form . From the initial condition (78), we have . To guarantee the existence of the fractional derivative we choose . Let ; then (77) reduces to

where , , and .

Performing the th derivatives one can see that for . For , we have

which yields .

Also,

For , we have

which yields .

Following analogous steps, we have for Thus

which is the exact solution.

4. Conclusion

For fractional differential equations of order , it is common to obtain a series solution in the form The question is how to obtain the coefficients . Naturally, if the problem is of fractional order, the differentiation is also of fractional order. In this paper, we presented a new algorithm for obtaining a series solution for nonlinear multiterm fractional differential equations of Caputo type, where we overcome the use of fractional differentiation. We employed a transformation that allows us to use ordinary differentiation rather than fractional differentiation to recursively compute the coefficient of the series expansion. Then the terms of the series, , are computed sequentially using a closed form formula. We applied the new algorithm to several types of multiterm fractional differential equations, where accurate solutions as well as exact solutions in closed forms have been obtained. For one example it is noted that the current algorithm is more efficient than the ADM as it is more easier to apply and it produces the exact solution while the ADM does not. We have developed the new algorithm for two- and three-term fractional equations, while the idea can be extended to multiterm fractional equations of arbitrary order; obtaining a general formula in this case is not an easy task. The current algorithm can be modified to deal with the fractional multiterm time-diffusion equations.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors gratefully acknowledge the support of the UAE University under Grant COS/IRG-11/15.

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