A Novel (G‘/G)-Expansion Method and its Application to the Space- Time Fractional Symmetric Regularized Long Wave (SRLW) Equation

In this work, we use the fractional complex transformation which converts nonlinear fractional partial differential equation to nonlinear ordinary differential equation. A fractional novel (G‘/G)-expansion method is used to look for exact solutions of nonlinear evolution equation with the aid of symbolic computation. To check the validity of the method we choose the space-time fractional symmetric regularized long wave (SRLW) equation and as a result, many exact analytical solutions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. The performance of the method is reliable, useful and gives more new general exact solutions than the existing methods.


Introduction
Fractional differential equations are generalizations of classical differential equations of integer order. Fractional Calculus, which is the field of mathematical analysis dealing with the investigation and applications of integrals and derivatives of arbitrary order, has attracted in recent years a considerable interest in many disciplines. It has been found that the behavior of many physical systems can be more properly defined by using the fractional theory. In recent years, considerable interest in fractional differential equations has been stimulated due to their numerous applications in the areas of physics and engineering [1]. Many important phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are well described by differential equations of fractional order [2][3][4][5][6]. Although the fractional calculus was invented by Newton and Leibniz over three centuries ago, it only became a hot topic recently owing to the development of the computer and its exact description of many real-life problems. A physical interpretation of the fractional calculus was given in [7]. With the development of symbolic computation software, like-Maple, many numerical and analytical methods to search for exact solutions of NLEEs have attracted more attention. As a result, the researchers developed and established many methods, for example, Cole-Hopf transformation [8], Tanh-function method [9][10][11][12][13], Inverse scattering transform method [14], Hirota method [15], Backlund transform method [16], Variational iteration method [17,18], Exp-function method [19][20][21][22][23], Extended tanh-method [24][25][26], Homogeneous balance method [27,28] and F-expansion method [29,30] are used for searching the exact solutions. Lately, Wang et al. [31] introduced a direct and concise method, called (G'/G)-expansion method and demonstrated that it is a powerful method for seeking analytic solutions of NLEEs. For additional references see the articles [32][33][34][35][36][37]. In order to establish the efficiency and diligence of (G'/G)-expansion method and to extend the range of applicability, further research has been carried out by several researchers. For instance, Zhang et al. [38] made a generalization of (G'/G)expansion method for the evolution equations with variable coefficients. Zhang et al. [39] also presented an improved (G'/G)-expansion method to seek more general traveling wave solutions. The (G'/G)-expansion method and the transformed rational function method used by Ma [40,41] have a common idea. That is, we firstly put the given NLEE into the corresponding ordinary differential equation (ODE), and then ODE can be transformed into a system of algebraic polynomials with the determining constants. By the solutions of the ordinary differential equation, we can obtain the exact traveling solutions and rational solution of the nonlinear evolution quations. In this article, we will apply novel (G'/G)-expansion method introduced by Alam et al. [42] to solve the space-time fractional partial differential equation in the sense of modified Riemann-Liouville derivative by Jumarie [43]. To illustrate the originality, consistency and advantages of the method, we will apply it to the space-time fractional SRLW equation and abundant new families of exact solutions are found. The Jumarie's modified Riemann-Liouville derivative [43] of order α is defined by the following expression: (1) Some important properties of Jumarie's derivative are:

Description of the Method
Suppose that a fractional partial differential equation in the independent variables say, x and t is given by (5) The main steps of the method are as follows: Step 1: Li et al. [44] proposed a fractional complex transformation to convert fractional partial differential equations into ordinary differential equations (ODE), so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus. The traveling wave variable (6)   (7) where the superscripts stand for the ordinary derivatives with respect to ζ Step 2. Integrate Eq. (7) term by term one or more times if possible, yields constant(s) of integration which can be calculated later.
Step 4: The value of the positive integer m can be determined by balancing the highest order linear terms with the nonlinear terms of the highest order come out in Eq. (7).
Step 6: Suppose the value of the constants can be obtained by solving the algebraic equations obtained in step 4. The values of the constants together with the solutions of Eq. (10) yield abundant exact traveling wave solutions of the nonlinear evolution equation (5).

Application of the Method to the Space-Time Fractional SRLW Equation
We consider the following space-time fractional SRLW equation (12) which arises in several physical applications including ion sound waves in plasma. This equation is symmetrical with respect to x and t. It arises in many nonlinear problems of mathematical physics and applied mathematics. Periodic wave solutions of SRLW have been given by using the Exp function method [46] and (G'/G)-expansion method [47]. Alzaidy [48] applied fractional subequation method to space-time fractional symmetric regularized long wave (SRLW) equation and find hyperbolic function solutions, trigonometric function solutions, and rational solutions. By the use of Eq. (4), Eq. (12) is converted into an ordinary differential equation and after integrating twice, we obtain (20)

Conclusions
A novel (G'/G)-expansion method is applied to fractional partial differential equation successfully. As applications, abundant new exact solutions for space-time fractional symmetric regularized long wave (SRLW) equation have been successfully obtained. The nonlinear fractional complex transformation for ζ is very important, which ensures that a certain fractional partial differential equation can be converted into another ordinary differential equation of integer order. The obtained solutions are more general, and many known solutions are only a special case of them. Thus novel (G'/G)-expansion method would be a powerful mathematical tool for solving nonlinear evolution equations.