Travelling Wave Solutions of Coupled Burger’s Equations of Time-Space Fractional Order by Novel (G/G)-Expansion Method

A R T I C L E I N F O A B S T R A C T Article history: Received: 09 March, 2017 Accepted: 29 March, 2017 Online: 07 April, 2017 In this paper, Novel (Gʹ/G)-expansion method is used to find new generalized exact travelling wave solutions of fractional order coupled Burger’s equations in terms of trigonometric functions, rational functions and hyperbolic functions with arbitrary parameters. For the conversion of the partial differential equation to the ordinary differential equation, complex transformation method is used. Novel (Gʹ/G)-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear equations. Moreover, for the representation of these exact solutions we have plotted graphs for different values of parameters which were in travelling waveform.


Introduction
Fractional complex transformation as in [1] is utilized for transformation of nonlinear fractional order partial differential equations into nonlinear ordinary differential equations. The onedimensional case of Burger's equation was introduced by a Dutch scientist J. M. Burger in 1939 see [2], its general form is given as ( ) + ( . ) = ∇ 2 . Afterward, new type of Burger's equation was presented in further research named as time-space fractional order coupled Burger's equations [3], which has the form where 0 < ≤ 1, 0 < ≤ 1, > 0.
An assortment of approaches occurs for nonlinear equations which gives their travelling wave and numerical solutions. Wang et al. introduced (Gʹ/G)-expansion method [4]. This method was further reached out for coupled Burger's equations by Younas et al. [3]. In this article Novel (Gʹ/G)-expansion method has been utilized to discover travelling wave solutions of nonlinear space and time fractional order coupled Burger's equations.

Fractional complex transformation
Fractional complex transform is the unobtrusive way for the transformation of the fractional order differential equation into integer order differential equation. This simplifies the rest of the procedure towards a solution. Presently consider where L, M, N and O are constants.

Travelling wave solution
Travelling wave solution was used for transformation of partial differential equations into ordinary differential equations. For this purpose, we consolidate two independent variables into one independent variable known as travelling wave variable x [5].
Applying phase plan analysis technique by presenting the travelling wave coordinates as takes after 1. Define the travelling wave variable x as = − , ∈ and > 0 is the propagation speed of the wave or travelling wave speed. 2. Defining the travelling wave Ansatz [6] for ( , ) in the following form ( ) = ( − ) = ( , ) By the travelling wave Ansatz, we have

Fractional derivatives in Caputo sense
Fractional derivative presented in the Caputo sense of ∈ −1 [7], is defined as where L is constant.
Linear relationships of Caputo's derivative are where θ and φ are constants. This relation is likewise named as Leibnitz rule.

Novel (Gʹ/G)-Expansion Method
Fractional order partial differential equation of the form see [8], where demonstrates the modified form of Riemann-Liouville derivative presented by Jumarie [9], and S is a polynomial of unknown function ( , ) and its several nonlinear partial and fractional derivatives.
Step I: Fractional complex transformation projected by Li and He [10] was used in order to transform PDE into ODE. For required equations, complex transformation is where L and M are non-zero arbitrary constants. Complex transformation (2) converts fractional order partial differential (1) into ODE in integer order as Step II: Integrate (3) to possible extent. At that time submit the constants which are to be determined later.
Step III: Considering the solution of (3) and − can't be simultaneously zero. and are constants that will be determined later. = ( ) satisfies the nonlinear ordinary differential equation of second order where P, Q and R are genuine constants and the derivative represented by prime is with respect to .
Step IV: By the homogeneous balance the value of m can be calculated, where > 0.
Step V: Embedding (4) together with (5) and (6)  Assembling each coefficient of polynomials equivalent to zero an overdetermined set is acquired as algebraic equations comprising , , and M.
Step VI: Values of , , and M the constants can be obtained by solving set of the algebraic equations. Cnsequently the solutions of (6) together with the resulted values of the constants generate exact travelling wave solutions of the nonlinear (1).

Application of Novel (Gʹ/G)-Expansion Method
Consider the Coupled Burger's equations of time-space fractional order form as By using fractional derivatives, (1) and (2) are converted into ordinary differential equations of integer order. After integration, we get here 1 and 2 are the constants of integration. By considering now the homogeneous balance of w and ′ we get = 1.
There are twenty more solutions of z.
For convenience, other exact solutions are overlooked.

Conclusion
Novel (Gʹ/G)-expansion method is an effective method for finding exact solutions of the fractional order partial differential equation.
As an application, exact solutions have been all around got for time-space fractional order coupled Burger's equations. The fractional complex transformation used as a part of the exhibited work is very momentous. By utilizing this fractional transformation, fractional order partial differential equation can be converted into the integer order ordinary differential equation. Graphical portrayals insure that the required solutions are travelling wave solutions. Novel (Gʹ/G)-expansion method is an influential mathematical tool for solving the nonlinear partial differential equations.