Mathematical Problems in Engineering

Volume 2018, Article ID 3892691, 10 pages

https://doi.org/10.1155/2018/3892691

## Applications of Separation Variables Approach in Solving Time-Fractional PDEs

Department of Mathematics, Honghe University, Mengzi, Yunnan 661199, China

Correspondence should be addressed to Yinghui He; moc.361@70iuhgniyeh

Received 16 December 2017; Accepted 15 March 2018; Published 22 April 2018

Academic Editor: Chaudry M. Khalique

Copyright © 2018 Yinghui He and Yunmei Zhao. 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

Based on the homogenous balanced principle and subequation method, an improved separation variables function-expansion method is proposed to seek exact solutions of time-fractional nonlinear PDEs. This method is novel and meaningful without using Leibniz rule and chain rule of fractional derivative which have been proved to be incorrect. By using this method, we studied a nonlinear time-fractional PDE with diffusion term. Some general solutions are obtained which contain many arbitrary parameters. Solutions given in related reference are just our especial case. And we also obtained some new type of solutions.

#### 1. Introduction

It is well known that fractional-order models are more adequate than previously used integer-order models due to an exact description of nonlinear phenomena. Therefore, nonlinear fractional partial differential equations (nfPDEs) have attracted more and more attention. Most recently, FPDEs are increasingly used in mathematical modeling of fluid mechanics, biological and chemical processes, signal processing, and control systems, and they are also used in fractal and differential geometry, and so on(see [1–8] and their references cited). Many natural phenomena associated with real-time problems which depend on both time instant and the previous time history, especially, can be successfully modeled by time-fractional nonlinear partial differential equations. Of course, investigating solutions of nfPDEs plays an important role in a large number of research fields. Until now, there is no general method. Nevertheless, many powerful methods were used for solving fractional differential equations, such as adomian decomposition method [9], symmetry method [10–12], numerical method [13, 14], first integral method [15], the homotopy perturbation method [16], invariant subspace method [17], and fractional variational iteration method [18], and have achieved significant progress.

There are many definitions of fractional derivative. The fractional differential equations can be expressed in terms of different differential operators defined by Riemann-Liouville, Caputo, Weyl, and many others. Among these definitions, Caputo definition is most frequently used. Here, let us briefly review this definition of fractional derivative.

The Caputo fractional derivative of order is defined by the following expression: where . In this paper we will adopt Caputo fractional derivative definition to investigate exact solution of the following type of nonlinear time-fractional partial differential equation: where is Caputo differential operator, , and , , . Equation (2) denotes a series of nonlinear time-fractional PDEs; these models defined by (2) can be used to accurately describe nonlinear phenomena in connection with real-time problems, which not only depend on time instant but also depend on the previous time history.

Recently, Feng [19] introduced a fractional method for seeking traveling wave solutions of space-time-fractional partial differential equations under the following modified Riemann-Liouville derivative definition [20]:

The following properties for the modified Riemann-Liouville derivative are usually used:

Let us briefly review this method. Using traveling wave transformation and (4) and (5), they reduced the following fractional partial differential equation: which can be converted into the following fractional ordinary differential equation with respect to the variable :

Then, they suppose that the solution of (7) can be expressed by a polynomial in as follows: where satisfies the following fractional ordinary differential equation:

In order to find exact solutions of (9), a nonlinear fractional complex transformation is used. Then, (9) can be turned into the following second ordinary differential equation: The exact solutions of (10) are known.

Finally, substituting (8) into (7), equating each coefficient of this polynomial on to zero, they can obtain a large number of exact solutions of space-time fractional partial differential equations (6).

We noticed two problems as follows. At first, in a short communication [21], Tarasov proved that formula (4) and the chain rule (5) cannot be performed together for fractional derivatives of noninteger orders . Therefore, using traveling wave transformation , space-time fractional partial differential equations (6) cannot be converted into fractional ordinary differential equation (7). And, (9) cannot convert to (10) under transformation .

In addition, it should be noted that the Leibniz rule in the form cannot hold for fractional derivatives of order for sets of differentiable and nondifferentiable functions [22]. It can be proved easily by using a counterexample. Let where . Obviously we have Then, one has

As a result, we have the condition which must be performed if the Leibniz rule (11) holds. For example, if we make and take into account , then condition (15) can be represented in the formwhere . It is not difficult to find that (16) holds only if . That is to say, the Leibniz rule (11) does not hold with . Therefore, the following formula used in references is incorrect:

Therefore, when substituting (8) into (7), the left-hand side of (7) cannot be expressed by the polynomial in .

So, this fractional method is not reliable and the obtained results are incorrect. By the way, we should be more cautious when using this kind of methods based on chain rule (5) and Leibniz rule (11), like the subequation method, the -expansion method, the exp-function method, the functional variable method, the trial equation method, the simple equation method, and so forth. We cannot obtain the exact solutions of compound function type of time-fractional PDEs (2) as in the references. Encouraged by Rui’s work [23], we shall introduce an improved method based on the homogenous balanced principle; by using this improved method we shall investigate exact solutions of a series of nonlinear time-fractional PDEs formed as (2).

The rest of this paper is organized as follows. In Section 2, we will introduce the improved separation variable expansion method based on the homogenous balanced principle. In Section 3, by using this new method, we will investigate exact solutions of a nonlinear time-fractional PDE with diffusion term discussed in (2).

#### 2. Introduction of Improved Separation Variable Function-Expansion Method

Although the fractional chain rule (5) and Leibniz rule (11) do not hold, they do not affect the investigation of the exact solutions of nonlinear time-fractional PDE (2) since formulas (4) still hold.

*Remark 1. *In reference [23], Rui points out that Leibniz rule (11) still holds, which is incorrect. Actually, in our method, it does not matter that Leibniz rule (11) does not hold. Because we just need the following formula: which is easy to be proved by definition for Caputo fractional derivative (1) and for the modified Riemann-Liouville derivative (3).

In the following, we introduce main steps of improved function-expansion method of separation variable type as follows.

*Step 1. *According to the formulas (4) and (18) and characters of nonlinear time-fractional partial differential equation (2), we suppose that (2) has the following exact solutions of the separation variable type: where the function can be taken as power function , Mittag-Leffler function , or , is a positive integer, and are constants to be determined later. The function satisfies the following subequation: where are constants. Some solutions of (20) are listed as follows.

When , where is an arbitrary integral constant.

When , When ,

Taking integral constant , under some conditions, we can obtain many special solutions of (20) which are listed in Table 1.