Trigonometric Function Solutions of Fractional Drinfeld ' s Sokolov-Wilson System

In this paper, we construct exact trigonometric solutions of the space-time fractional classical Drinfeld's Sokolov-Wilson system by Modified Trial Equation Method (MTEM). These solutions may explain some physical phenomena and lead to researchers in physics and engineering.


Introduction
Studies in recent years on fractional analysis have been increasingly proceed.Fractional analysis describe many scientific phenomena such as physics, chemistry, biology, engineering and so on.Several efficient analytical, semi-analytical and numerical methods are applied for solving fractional equations and equation systems.Some of them are Modified Trial Equation method, Sumudu Transform method, Riccati-Bernoulli sub-ODE method, İmproved Bernoulli sub-equation method, modified Kudryashov method, Functional Variable method, Homotopy Analysis method and so on have been used to find new solutions of fractional equations [1][2][3][4][5][6][7][8][9][10][11] We apply Modified Trial Equation method (MTEM) for obtain new travelling wave solution to fractional Drinfeld's Sokolov-Wilson system [1,12].
We assume that the solution can be expressed in the form where ( ) F u and ( ) G u are polynomials.Substituting above relations into Eq.(2.3) yields an equation of polynomial ( ) u According to the balance principle, we can get a relationship between n and l .We can compute some values of n and l .
Using a complete discrimination system for polynomial of ( ) F u , we solve Eq.(2.8) with the help of Mathematica 9 and classify the exact solutions to Eq.(2.3).For better explication of results obtained in this way, we can plot two and three dimensional surfaces of the solutions obtained by using suitable parameters.

Application
Let consider the the space-time fractional classical Drinfeld's Sokolov-Wilson system as follows [1], where and u v are the functions of ( , ) We apply following transformations then the Eq.(3.1) can be reduced to ordinary differential equation.
where w and  are arbitrary constants and We obtain following equalities from Eq.(3.3), , paw U VV b where 1 c is constant of integration.Embedding Eqs.(3.5,3.6)into Eq.(3.4), we have the nonlinear ordinary differential equation: Integrating Eq.(3.7) once, where 1 c is constant of integration.When we reconsider the Eq.(3.8) for homogenous balance principle between and , we obtain the following relationship for n and l , (3.9) Case 1: For the values of , we get ; we have coefficients above.By substituting these coefficients in Eq.(3.8), we obtain the solution of ( , ) v x t and ( , )

Conclusions
In this manuscript, we have efficiently and easily practice the Modified Trial Equation Method that is give analytical solution.We can say that this method is applied to nonlinear differential equations and systems.These trigonometric function solutions have been introduced to the literature for the first time.We think that these new solutions lead the way other scientific area.
Reduce Eq.(2.4) to the elementary integral form, the algebraic equation system is solved,