The Improved exp ( − φ ( ξ ) ) Fractional Expansion Method and its Application to Nonlinear Fractional Sharma-Tasso-Olver Equation

In this paper, we propose a new method called exp(−φ(ξ)) fractional expansion method to seek traveling wave solutions of the nonlinear fractional Sharma-Tasso-Olver equation. The result reveals that the method together with the new fractional ordinary differential equation is a very influential and effective tool for solving nonlinear fractional partial differential equations in mathematical physics and engineering. The obtained solutions have been articulated by the hyperbolic functions, trigonometric functions and rational functions with arbitrary constants. Citation: Alhakim LA, Moussa AA (2017) The Improved exp(−φ(ξ)) Fractional Expansion Method and its Application to Nonlinear Fractional SharmaTasso-Olver Equation. J Appl Computat Math 6: 360. doi: 10.4172/2168-9679.1000360


Introduction
It is well known that nonlinear fractional partial differential equations (NFPDEs) are widely used as models to describe many important complex physical phenomena in various fields of science, such as plasma physics, nonlinear optics, solid state physics, fluid mechanics, fluid flow, chemical kinematics, chemistry, biology, finance, economy, and so on. Thus, establishing exact traveling wave solutions of NFPDEs is very important to better understand nonlinear phenomena's as well as other real-life applications.
In the past, a wide range of methods have been developed to generate analytical solutions of nonlinear partial differential equations.
In recent years, several attempts have succeeded in the synthesis of the previous methods to searching for exact solutions to nonlinear fractional differential equations. Zhang and Zhang [12][13][14] proposed on the basis of homogeneous balance principle and Jumarie's modified Riemann-Liouville derivative a new direct method called fractional sub-equation method to search for explicit solutions of nonlinear time fractional biological population model and (4+1) dimensional space-time fractional Fokas equation. Wangi and Xu [15] improved this method to obtain the exact solutions of the space-time fractional generalized Hirota-Satsuma coupled Korteweg-de Vries equations.
The remainder of the paper is organized as follows. Section 2 gives some definitions and properties of the modified Riemann-Liouville derivative [16], and explains the improved exp (−ϕ(ξ)) fractional expansion method. Section 3 applies this method for solving the nonlinear fractional Sharma-Tasso-Olver equation. Section 4 concludes the paper.

Jumarie's Modified Riemann-Liouville Derivative and the Improved exp(−ϕ(ξ))Fractional Expansion Method
In this section, we briefly review the main definitions and properties of the fractional calculus proposed by Jumarie [17] which will be used in the following section.
The modified Riemann-Liouville derivative as defined by Jumarie [18] is: Some useful formulas and properties of Jumarie's modified Riemann-Liouville derivative were summarized in [18], among them the three following formulas: (1 ) , Now, we outline the main steps of the exp(−ϕ(ξ)) fractional expansion method to solve fractional differential equations. Suppose that a fractional partial differential equation, say in the independent variables x and t, is given by Where u=u(x,t) is an unknown function, F is a polynomial in are the modified Riemann Liouville derivatives of u with respect to t and x, respectively. The main steps of this method are as follows: Step 1: Use the traveling wave transformation: Where k is a non-zero constant to be determined latter, which reduces (2.5) to an (NFODE) for u=u(ξ) in the form: Step 2: Balance the highest derivative term with the nonlinear terms in (2.7) to find the value of the positive integer (m). If the value (m) is non-integer one can transform the equation studied.
Step 5: Suppose that the value of the constants k, λ,µ and α i (i=−m,...,m) can be found by solving the algebraic equations which are obtained in Step 4. Since the general solutions of (2.9) have been well known, substituting k,λ,µ,α i and the solutions of (2.9) into (2.8), we obtain the exact solutions for eqn. (2.5).

Conclusion
In this article, we proposed a new method called improved exp (−ϕ(ξ)) fractional expansion method using the generalized wave transformation (2.6) and the auxiliary fractional differential equation (2.9), to obtain the exact solutions of nonlinear fractional Sharma-Tasso-Olver equation. The main advantage of this method is its capability of greatly reducing the size of computational work compared to existing techniques. The method could be used for a large class of very interesting nonlinear equations. These solutions have rich local structures, it may be important to explain some physical phenomena. This work shows that, the improved exp (−ϕ(ξ)) fractional expansion method is direct, effective and can be used for many other FNLPDEs in mathematical physics.