Explicit solutions to the Sharma-Tasso-Olver equation

We present new exact traveling wave solutions of generalized Sharma-Tasso-Olver (STO) with variable coefficients using three different methods, namely the extended F-expansion, the new sub-equations, and generalized Kudryashov expansion. We obtain new solutions with the form of solitons, triangular and rational functions. Computational results indicate that these methods are very useful and easily applicable for solving diverse types of differential equations in nonlinear science.


Introduction
Research on exact solutions of nonlinear differential equations with variable coefficients has been a significant area for recent decades, see e.g. [1−25]. We consider nonlinear STO equation with variable coefficients [26,27].
In order to study the traveling wave propagation solution of STO [43,44], let us introduce: in which α is a parameter and ω is wave speed. By Eq (1.2), Eq (1.1) is written in which f (t)and g (t) satisfy f (t) = 3g (t). Integrating Eq (1.3), we get In this study we get solitary wave and the periodic wave solutions by using algebraic direct method, Sub-equations method and F-expansion method. In the next two sections, the new proposed methods are presented and different types of exact solutions of STO are written down. Section 4 is devoted to the conclusion.

Methodology
Let Z be a polynomial function of x, and t. Consider the nonlinear PDE

Extended F-Expansion method
Let the solution be written as in which a 0 and a i are constants, and M 0 is a natural number and χ (ζ)satisfies where, χ (ζ) = dχ dζ and A, B, C are parameters. In order to solve Eq (1.4) via F−expansion method, equating u ξξ with u 3 yields M = 1. Hence, Eq (2.4) reads in which a 0 , a 1 and a −1 are constants. Inserting Eq (2.6) into the reduced Eq (1.4) yields: Case (1.1): a −1 = 0, a 1 = 1, a 0 = −1, α = α and ω = −α. Using the transformation (1.2), the corresponding solution in terms of the original coordinates is as follows where g (t) is an arbitrary function.
where g (t) is an arbitrary function.
where g (t) is an arbitrary function.
Using the transformation (1.2), the corresponding solution in terms of the original coordinates is taken as where g (t) is an arbitrary function.
where g (t) is an arbitrary function.

New sub-equation method
In view this method (MAE) [24], affirms the general solution as the form as The parametersa j , b j are arbitrary constants and f (ς)satisfy the following auxiliary equation in which α, β, σare arbitrary constants and A > 0, A 1.

Conclusions
Methods of the extended sub-equation, direct algebraic and F-expansion have been successfully applied to solve the variable coefficient STO equation with its fission and fusion. Using the F-expansion method, one may able to classify ten types of solutions in terms of the arbitrary function g (t). The advantage of the presence of that arbitrary function g (t), enable us to construct a wide range classes of solutions according to the different choices of g (t)and any initial condition may be persuaded.
On the other hand, using different mathematical methods may lead us to another type of solutions. For example, applying the improved tanh method, one obtains the following type of solution u (x, t) = ± sec x + α t 0 g t dt ± tan x + α t 0 g t dt , (4.1) that maps to the triangular periodic solution where ω = α. In addition, one may also obtain the numerous soliton like solutions, (4.2) where ω = −α and g (t) is an arbitrary function of t. Application of these methods to fractal order PDEs may be seen in, e.g. [25][26][27][45][46][47][48][49][50][51][52][53]. We will investigate the applicability of these methods to fractional stochastic differential equations in a future work.