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Journal of Applied Mathematics
Volume 2013, Article ID 481729, 10 pages
http://dx.doi.org/10.1155/2013/481729
Research Article

A New Fractional Subequation Method and Its Applications for Space-Time Fractional Partial Differential Equations

1School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
2School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 22 February 2013; Accepted 1 July 2013

Academic Editor: Magdy A. Ezzat

Copyright © 2013 Fanwei Meng and Qinghua Feng. 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

A new fractional subequation method is proposed for finding exact solutions for fractional partial differential equations (FPDEs). The fractional derivative is defined in the sense of modified Riemann-Liouville derivative. As applications, abundant exact solutions including solitary wave solutions as well as periodic wave solutions for the space-time fractional generalized Hirota-Satsuma coupled KdV equations are obtained by using this method.

1. Introduction

Fractional differential equations are generalizations of classical differential equations of integer order. Recently, fractional differential equations have been the focus of many studies due to their frequent appearance in various applications in physics, biology, engineering, signal processing, systems identification, control theory, finance, and fractional dynamics. Among the investigations for fractional differential equations, research for seeking exact solutions and approximate solutions of fractional differential equations is a hot topic. New exact solutions for fractional differential equations may help to understand better corresponding nonlinear wave phenomena they describe. Some powerful methods have been proposed so far (e.g., see [112]). Using these methods, a variety of fractional differential equations have been investigated.

In this paper, we propose a new fractional subequation method to establish exact solutions for fractional partial differential equations (FPDEs), which is based on the following fractional ordinary differential equation: wheredenotes the modified Riemann-Liouville derivative of orderforwith respect to.

The definition and some important properties for Jumarie’s modified Riemann-Liouville derivative of orderare listed as follows (see [1316]):

We organize this paper as follows. In Section 2, we derive the expression forrelated to (1). In Section 3, we give the description of the fractional subequation method for solving FPDEs. Then in Section 4 we apply this method to establish exact solutions for the space-time fractional generalized Hirota-Satsuma coupled KdV equations. Some conclusions are presented at the end of the paper.

2. The General Expression for

In order to obtain the general solutions for (1), we suppose and a nonlinear fractional complex transformation. Then, by (3) and the first equality in (5), (1) can be turned into the following second ordinary differential equation: By the general solutions of (6), we havewhere and are arbitrary constants.

Since, we obtain

3. Description of the Fractional Subequation Method

In this section, we give the main steps of the fractional subequation method for finding exact solutions for FPDEs.

Suppose that an FPDE, say in the independent variables, is given by where, , are unknown functions and is a polynomial in and their various partial derivatives including fractional derivatives.

Step 1. Suppose that Then, by the second equality in (5), (9) can be turned into the following fractional ordinary differential equation with respect to the variable:

Step 2. Suppose that the solution of (11) can be expressed by a polynomial inas follows: wheresatisfies (1),is a constant, and , , , are constants to be determined later. The positive integer can be determined by considering the homogeneous balance between the highest-order derivatives and nonlinear terms appearing in (11).

Step 3. Substituting (12) into (11), using (1), and collecting all terms with the same order oftogether, the left-hand side of (11) is converted into another polynomial in. Equating each coefficient of this polynomial to zero yields a set of algebraic equations for, , .

Step 4. Solving the equations in Step 3 and using (8), we can construct a variety of exact solutions for (9).

4. Application of the Method to Space-Time Fractional Generalized Hirota-Satsuma Coupled KdV Equations

In this section, we will apply the described method in Section 3 to solve the space-time fractional generalized Hirota-Satsuma coupled KdV equations [15, 16]: Equations (13) can be used to describe the interaction of two long waves with different dispersion relations [17]. In [15], the authors solved equations (13) by a proposed fractional subequation method based on the fractional Riccati equation, while in [16], (13) are solved by the known ()-expansion method. Now we apply the described method in Section 3 to solve (13). To begin with, suppose that , , where, are all constants with. Then, by usinge the second equality in (4), we obtain and similarly we have then (11) can be turned into the following fractional ordinary differential equations with respect to the variable:

Suppose that the solutions of (16) can be expressed by Balancing the order of and , and , and and in (16), we have  . So

Substituting (18) into (16), using (1), and collecting all the terms with the same power oftogether, equating each coefficient to zero yields a set of algebraic equations. Solving these equations, with the aid of the mathematical software Maple, yields the following seven groups of values.

Case 1. One has

Case 2. One has

Case 3. One has

Case 4. One has

Case 5. One has

Case 6. One has

Case 7. One has
Substituting the previous results into (18) and combining with (8), we can obtain a series of exact solutions for (13).

From Case 1, we obtain the following exact solutions.

When, where.

When, where and  .

In particular, if we letin (26)–(28), then we obtain the following solitary wave solutions, which are shown in Figures 1, 2, and 3:

481729.fig.001
Figure 1: The solution in (32) with , , , .
481729.fig.002
Figure 2: The solution in (33) with , , , , .
481729.fig.003
Figure 3: The solution in (34) with , , , , .

If we letin (29)–(31), then we obtain the following periodic wave solutions, which are shown in Figures 4, 5, and 6:

481729.fig.004
Figure 4: The solution in (35) with , , , .
481729.fig.005
Figure 5: The solution in (36) with , , , .
481729.fig.006
Figure 6: The solution in (37) with , , , .

Similar to the established solutions from Case 1, we can construct corresponding exact solutions to (13) from Cases 27, which are omitted here.

Remark 1. We note that the solutions obtained here are of new forms compared with the solutions obtained in [15, 16] since a fully new method is used here.

5. Conclusions

Based on the concept of the modified Riemann-Liouville derivative and a variable transformation, we have proposed a new fractional subequation method for solving fractional partial differential equations (FPDEs). By using this method, the space-time fractional generalized Hirota-Satsuma coupled KdV equations are solved successfully, and, as a result, some exact solutions are established, which may help to understand better the nonlinear wave phenomena. It is supposed that this method can be further applied to solve other FPDEs.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (11171178) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province. The authors would like to thank the reviewers very much for their valuable suggestions in the paper.

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