Advances in Mathematical Physics

Volume 2015, Article ID 567842, 10 pages

http://dx.doi.org/10.1155/2015/567842

## Exact Solutions for Some Fractional Differential Equations

Department of Mathematics, Faculty of Science and Arts, Bozok University, 66100 Yozgat, Turkey

Received 10 February 2015; Accepted 29 April 2015

Academic Editor: Andrei D. Mironov

Copyright © 2015 Abdullah Sonmezoglu. 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

The extended Jacobi elliptic function expansion method is used for solving fractional differential equations in the sense of Jumarie’s modified Riemann-Liouville derivative. By means of this approach, a few fractional differential equations are successfully solved. As a result, some new Jacobi elliptic function solutions including solitary wave solutions and trigonometric function solutions are established. The proposed method can also be applied to other fractional differential equations.

#### 1. Introduction

Fractional differential equations attracted attention in physics, biology, engineering, signal processing, systems identification, control theory, finance, and fractional dynamics [1–3]. Also, they are employed in social sciences such as food supplement, climate, finance, and economics.

Finding approximate and exact solutions to fractional differential equations is an important task. Various analytical and numerical methods have been introduced to obtain solutions of fractional differential equations, such as the Adomian decomposition method [4, 5], the variational iteration method [6–8], the homotopy analysis method [9–12], the homotopy perturbation method [13–15], the Lagrange characteristic method [16], the finite difference method [17], the finite element method [18], the differential transformation method [19], the fractional subequation method [20–24], the first integral method [25], the -expansion method [26–29], the fractional complex transform method [30], and the modified simple equation method [31–33].

In [34], Jumarie proposed a modified Riemann-Liouville derivative. With this kind of fractional derivative and some useful formulas, we can convert fractional differential equations into integer-order differential equations by variable transformation.

In this paper, we used extended Jacobi elliptic function expansion method [35–37] to establish exact solutions for three nonlinear space-time fractional differential equations in the sense of Jumarie’s modified Riemann-Liouville derivative, namely, the space-time fractional generalized reaction duffing equation, the space-time fractional bidirectional wave equations, and the space-time fractional symmetric regularized long wave (SRLW) equation. Also, we included figures to show the properties of some Jacobi elliptic function solutions of these fractional differential equations.

#### 2. Jumarie’s Modified Riemann-Liouville Derivative and the Extended Jacobi Elliptic Function Expansion Method

In this section, we first give the definition and some properties of the modified Riemann-Liouville derivative which are used further in this paper.

The Jumarie modified Riemann-Liouville derivative of order is defined by the expression [34]where , denote a continuous (but not necessarily differentiable) function.

Some properties of the fractional modified Riemann-Liouville derivative were summarized and three useful formulas of them are [34]

Next, let us consider nonlinear partial fractional differential equationwhere is an unknown function and is a polynomial of . In this equation, the partial fractional derivatives involving the highest order derivatives and the nonlinear terms are included.

Li and He [38] presented a fractional complex transform to convert fractional differential equations into ordinary differential equations (ODEs), so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus. By using the traveling wave variablewhere , are nonzero arbitrary constants and is the wave speed, we can rewrite (3) as the following nonlinear ODE:where the prime denotes the derivation with respect to . If possible, we should integrate (5) term by term one or more times.

Our main goal is to derive exact or at least approximate solutions, if possible, for this ODE. For this purpose, using the extended Jacobi elliptic function expansion method, can be expressed as a finite series of Jacobi elliptic functions, , that is, the ansatz:The parameter is determined by balancing the linear term(s) of highest order with the nonlinear one(s). Andwhere and are the Jacobi elliptic cosine function and the Jacobi elliptic function of the third kind, respectively, with the modulus . Therefore, the highest degree of is taken asSubstituting (6)–(8) into (5) and comparing the coefficients of each power of in both sides, we get an overdetermined system of nonlinear algebraic equations with respect to , , and . Solving this system, with the aid of Mathematica, then , , and can be determined. Substituting these results into (6), then some new Jacobi elliptic function solutions of (3) can be obtained. We can get other kinds of Jacobi doubly periodic wave solutions.

Since degenerates, respectively, as the following form. (1)Solitary wave solutions:(2)Triangular function formal solution:

#### 3. Applications of the Method

In this section, we present three examples to demonstrate the effectiveness of our approach to solve nonlinear fractional partial differential equations.

##### 3.1. Space-Time Fractional Generalized Reaction Duffing Equation

We have applied the extended Jacobi elliptic function expansion method to construct the exact solutions of space-time fractional generalized reaction duffing equation [39, 40] in the formwhere , , , and are all constants. Equation (12) reduces many well-known nonlinear fractional wave equations such as the following.(i)Fractional Klein-Gordon equation:(ii)Fractional Landau-Ginzburg-Higgs equation:(iii)Fractional equation:(iv)Fractional duffing equation:(v)Fractional Sine-Gordon equation:For our purpose, we introduce the following transformations:where is a wave variable and and are constants; all of them are to be determined. Substituting (18) into (12), (12) is reduced into an ODE:where . Suppose that the solution of (19) can be expressed byConsidering the homogeneous balance between the highest order derivative and the highest order nonlinear term in (19), we obtain . SoSubstituting (21) into (19) and comparing the coefficients of each power of in both sides, we get an overdetermined system of nonlinear algebraic equations with respect to , , , and . Solving this system with Mathematica, we get the following results.

*Case 1. *Consider

*Case 2. *Consider

*Case 3. *Consider

Thus, we obtain the following solutions of (12).

*Solution 1. *See Figure 1: