Exp-Function Method for Solving Fractional Partial Differential Equations

We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.


Introduction
Nonlinear differential equations of integer order (NLDEs) can be used to describe many nonlinear phenomena such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, and chemical physics. In the research of the theory of NLDEs, searching for more explicit exact solutions to NLDEs is one of the most fundamental and significant studies in recent years. With the help of computerized symbolic computation, much work has focused on the various extensions and applications of the known algebraic methods to construct the solutions to NLDEs. There have been a variety of powerful methods. For example, these methods include the generalized Riccati subequation method [1,2], the Jacobi elliptic function expansion [3,4], theexpansion method [5], the Exp-function method [6][7][8][9], and the ( / )-expansion method [10,11].
Fractional differential equations are generalizations of classical differential equations of integer order and have recently proved to be valuable tools to the modeling of many physical phenomena and 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. In order to obtain exact solutions for fractional differential equations, many powerful and efficient methods have been proposed so far (e.g., see [12][13][14][15][16][17][18][19][20][21][22][23][24]). But nobody has researched the application of the Exp-function method to fractional differential equations so far to our best knowledge.
In this paper, we will apply the Exp-function method for solving fractional partial differential equations in the sense of modified Riemann-Liouville derivative by Jumarie [25]. By a certain nonlinear fractional complex transformation = ( , 1 , 2 , . . . , ), a given fractional partial differential equation expressed in independent variables , 1 , 2 , . . . , can be turned into another ordinary differential equation of integer order, whose solutions are supposed to have one of the following forms: where 1 , 2 , 1 , and 2 are unknown positive integers that will be determined by using the balance method. Determine the highest order nonlinear term and the linear term of highest order in the ordinary differential equation and express them in terms of (3). Then, in the resulting terms, balance the highest order Exp-function to determine 2 and 2 and balance the lowest order Exp-function to determine 1 and 2 The Scientific World Journal The Jumarie's modified Riemann-Liouville derivative of order is defined by the following expression [25]: We list some important properties for the modified Riemann-Liouville derivative as follows [25, (3.10)-(3.13)] (see also in [22][23][24]26]): In the following, we will apply the Exp-function method to find exact solutions for the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation.
Substituting (10) into (7), eliminating the denominator, 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 with the aid of the mathematical software Maple yields the following family of values of , , = −1, 0, 1.

Family 1. Consider
where 1 , 0 are arbitrary constants. Substituting the result above into (10), we can obtain the following exact solutions to (7): The Scientific World Journal 3 0 = 2 1 , then we obtain the following hyperbolic function solitary wave solution: Case 2. Let 1 = 1 = , 2 = 2 = , and for simplicity, we take Substituting (14) into (7), eliminating the denominator, 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 the following result.
Substituting (19) into (7), eliminating the denominator, 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 the following.

Nonlinear Fractional Sharma-Tasso-Olver (STO) Equation.
We consider the nonlinear fractional Sharma-Tasso-Olver (STO) equation with time-fractional derivative [26,27]: In [26], the author solved (27) by the first integral method and obtained some hyperbolic function and trigonometric function solutions, while in [27], the variational iteration method, the Adomian decomposition method, and the homotopy perturbation method are used for obtaining a rational approximation solution for (27). Now we will apply the Exp-function method to (27). To begin with, we suppose ( , ) = ( ), where = + ( /Γ(1 + )) + 0 , , , 0 are all constants with , ̸ = 0. Then by use of (5), (27) can be turned into First we suppose that the solution of (28) can be expressed by Similar to the previous subsection, by balancing the highest order Exp-function in and 2 , we have 2 = 2 , while we obtain 1 = 1 by balancing the lowest order Expfunction in and 2 . Similar to the solving process for the space-time fractional Fokas equation, we will also proceed the computation under two selected cases.
Substituting (30) into (28), eliminating the denominator, 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 two families of values for , , = −1, 0, 1.

Family 1. Consider
The Scientific World Journal 5 where 1 , −1 are arbitrary constants.

Case 2. Let
Substituting (36) into (28), eliminating the denominator, 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 the following results.
Substituting (43) into (28), eliminating the denominator, 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, we obtain the following two families of values.
Family 6. Consider where 1 , 0 , −1 are arbitrary constants. Substituting the results above into (43), we can obtain the following exact solutions to (28): If we especially take = − 3 , 1 = −1 or = − 3 , (46), then we obtain the trigonometric function solutions: Case 2. Let Substituting (49) into (28), eliminating the denominator, 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, we obtain the following two families of values.
The Scientific World Journal 7 Substituting the results above into (49), we can obtain the following exact solutions to (28): 3 )

Conclusions
We have extended the Exp-function method to solve fractional partial differential equations successfully. As applications, some generalized and new exact solutions for the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation have been successfully found. As one can see, this method is based on the homogeneous balancing principle. So it can also be applied to other fractional partial differential equations where the homogeneous balancing principle is satisfied.