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

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 ofmodified 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.


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 [1][2][3][4][5][6][7][8][9][10][11][12]).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:  2    () +     () +  () = 0, 0 <  ≤ 1, (1) where    () denotes the modified Riemann-Liouville derivative of order  for () with respect to .
The definition and some important properties for Jumarie's modified Riemann-Liouville derivative of order  are listed as follows (see [13][14][15][16]): We organize this paper as follows.In Section 2, we derive the expression for    ()/() related 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.

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 , where   =   (,  1 ,  2 , . . .,   ),  = 1, . . ., , 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 in (   /) as follows: where  = () satisfies ( 1),  is a constant, and  , ,  = 0, 1, . . ., ,  = 1, 2, . . ., , 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 4. Solving the equations in Step 3 and using (8), we can construct a variety of exact solutions for (9).

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 (, ) = (), V(, ) = (), (, ) = (), where  =  +  +  0 , , ,  0 are all constants with ,  ̸ = 0.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  ). (18) Substituting ( 18) into ( 16), using (1), and collecting all the terms with the same power of (   /) √(1 + (1/)(   /) 2 ) together, 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 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  < 0, where When  > 0, where  =  +  +  0 and  = (1/4) −4  2 2 .In particular, if we let  2 = 0 in (26)-( 28), then we obtain the following solitary wave solutions, which are shown in Figures 1, 2, and 3: If we let  2 = 0 in (29)-(31), then we obtain the following periodic wave solutions, which are shown in Figures 4, 5, and  6: Similar to the established solutions from Case 1, we can construct corresponding exact solutions to (13) from Cases 2-7, 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.

Conclusions
Based on the concept of the modified Riemann-Liouville derivative and a variable transformation  =  +  1  1 +  2  2 + ⋅ ⋅ ⋅ +     +  0 , 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.