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Exact solutions for fractional partial differential equations by a new fractional sub-equation method
Advances in Difference Equations volume 2013, Article number: 199 (2013)
In this paper, we propose a new fractional sub-equation method for finding exact solutions of fractional partial differential equations (FPDEs) in the sense of modified Riemann-Liouville derivative, which is the fractional version of the known (G′/G) method. To illustrate the validity of this method, we apply it to the space-time fractional Fokas equation, the space-time fractional -dimensional dispersive long wave equations and the space-time fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established.
Fractional differential equations are generalizations of classical differential equations of integer order. In recent decades, 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. Many articles have investigated some aspects of fractional differential equations such as the existence and uniqueness of solutions to Cauchy-type problems, the methods for explicit and numerical solutions, and the stability of solutions [1–8]. In , Jafari et al. applied the fractional sub-equation method to construct exact solutions of the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation. In , Baleanu et al. studied the existence and uniqueness of the solution for a nonlinear fractional differential equation boundary-value problem by using fixed-point methods. In , Nyamoradi et al. investigated the existence of solutions for the multipoint boundary value problem of a fractional order differential inclusion.
Among the investigations for fractional differential equations, research into seeking exact solutions and numerical solutions of fractional differential equations is an important topic. Many powerful and efficient methods have been proposed to obtain numerical solutions and exact solutions of fractional differential equations so far. For example, these methods include the Adomian decomposition method [12–14], the variational iterative method [15–22], the homotopy perturbation method [23–26], the differential transformation method , the finite difference method , the finite element method , the fractional Riccati sub-equation method [30–32] and so on. In these investigations, we note that many authors have sought exact and numerical solutions for fractional partial differential equations (FPDEs) in the sense of modified Riemann-Liouville derivative (for example, see [16, 17, 30–34]). Based on these methods, a variety of fractional differential equations have been investigated.
In this paper, we propose a new fractional sub-equation method to establish exact solutions for fractional partial differential equations (FPDEs) in the sense of modified Riemann-Liouville derivative defined by Jumarie , which is a fractional version of the known (G′/G) method [36–39]. This method is based on the following fractional ODE:
where denotes the modified Riemann-Liouville derivative of order α for with respect to ξ.
The rest of this paper is organized as follows. In Section 2, we present some definitions and properties of Jumarie’s modified Riemann-Liouville derivative and the expression for related to Eq. (1). In Section 3, we give the description of the fractional sub-equation method for solving FPDEs. Then in Section 4 we apply this method to establish exact solutions for the space-time fractional Fokas equation, the space-time fractional -dimensional dispersive long wave equations and the space-time fractional fifth-order Sawada-Kotera equation. Some conclusions are presented at the end of the paper.
2 Jumarie’s modified Riemann-Liouville derivative and general expression for
Jumarie’s modified Riemann-Liouville derivative of order α is defined by the following expression :
We list some important properties for the modified Riemann-Liouville derivative as follows (see [, Eqs. (3.10)-(3.13)]):
In order to obtain the general solutions for Eq. (1), we suppose and a nonlinear fractional complex transformation . Then by Eq. (2) and the first equality in Eq. (4), Eq. (1) can be turned into the following second ordinary differential equation
By the general solutions of Eq. (5), we have
where , are arbitrary constants.
Since , we obtain
3 Description of the fractional sub-equation method
In this section we describe the main steps of the fractional sub-equation method for finding exact solutions of FPDEs.
Suppose that a fractional partial differential equation, say in the independent variables , is given by
where , , are unknown functions, P is a polynomial in and their various partial derivatives include fractional derivatives.
Step 1. Suppose that
Step 2. Suppose that the solution of (10) can be expressed by a polynomial in as follows:
where satisfies Eq. (1), and , , , are constants to be determined later with . The positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (10).
Step 3. Substituting (11) into (10) and using (1), collecting all terms with the same order of together, the left-hand side of (10) 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 equation system in Step 3 and using (7), we can construct a variety of exact solutions for Eq. (8).
Remark 1 If we set in Eq. (1), then it becomes , which is the foundation of the known (G′/G) method for solving partial differential equations (PDEs). So, in this way, the described fractional sub-equation method above is the extension of the (G′/G) method to fractional case.
Remark 2 The idea of the transformation from n independent variables to one independent variable denoted in Eq. (9) is similar to that in [, Eq. (12)], [, Eq. (8)], and [, Eq. (6)]. After applying this transformation to Eq. (8), by use of the second equality of Eq. (4), the original fractional partial differential equation can be transformed into another fractional ordinary differential equation in one independent variable.
4 Applications of the method
4.1 Space-time fractional Fokas equation
We consider the space-time fractional Fokas equation
Suppose that the solution of Eq. (13) can be expressed by
Substituting (15) into (13) and collecting all the terms with the same power of together, equating each coefficient to zero, yield a set of algebraic equations. Solving these equations yields
Remark 3 As one can see, the established solutions for the space-time fractional Fokas equation above are different from the results in  and are new exact solutions so far to our best knowledge.
Remark 4 The method used above in solving Eq. (12) can also be used to obtain solutions to initial or boundary value problems. For example, if we add the initial value condition
then, after substituting this initial value condition to the trigonometric solutions obtained in Eq. (17) and fulfilling some basic comparison and computation, one can see that , , , , , , , , . So, we obtain the solution to the initial value problem as
4.2 Space-time fractional -dimensional dispersive long wave equations
We consider the following space-time fractional -dimensional dispersive long wave equations
Some types of exact solutions for Eqs. (20) have been obtained in [40–53] by use of various methods including the Riccati sub-equation method [40, 41, 46], the nonlinear transformation method , the Jacobi function method [44, 45, 53], the (G′/G)-expansion method , the modified CK’s direct method , the EXP-function method , the Hopf-Cole transformation method , the modified extended Fan’s sub-equation method [50, 51], the generalized algebraic method . But we notice that so far no research has been pursued for Eqs. (19). In the following, we will apply the proposed fractional sub-equation method to Eqs. (19).
Suppose that the solution of Eqs. (21) can be expressed by
Balancing the order of and , and in (21), we can obtain , . So, we have
Substituting (23) into (21), using Eq. (1) and collecting all the terms with the same power of together, equating each coefficient to zero, yield a set of algebraic equations. Solving these equations yields
4.3 Space-time fractional fifth-order Sawada-Kotera equation
We consider the space-time fractional fifth-order Sawada-Kotera equation
which is the variation of the fifth-order Sawada-Kotera equation 
Suppose that the solution of Eq. (29) can be expressed by
Balancing the order of and in Eq. (30), we have . So,
Substituting (31) into (29), using Eq. (1) and collecting all the terms with the same power of together, equating each coefficient to zero, yield a set of algebraic equations. Solving these equations yields
We have proposed a new fractional sub-equation method for solving FPDEs successfully, which is the fractional version of the known (G′/G) method. As one can see, the two nonlinear fractional complex transformations for ξ and η used here are very important. The first transformation ensures that a certain fractional partial differential equation can be turned into another fractional ordinary differential equation, the solutions of which can be expressed by a polynomial in , where G satisfies the fractional ODE . The general expression for related to this fractional ODE can be obtained due to the second fractional complex transformations for η. Finally, we note that with this kind of nonlinear fractional complex transformations, it is worth to investigate the applications of other algebraic methods to fractional partial differential equations such as the Exp-function method, F-expansion method, Jacobi elliptic function method and so on.
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The authors would like to thank the reviewers very much for their valuable suggestions on the paper.
The authors declare that they have no competing interests.
BZ carried out the main part of this article. All authors read and approved the final manuscript.