Exact Solution to Nonlinear Differential Equations of Fractional Order via ( G ʹ / G )-Expansion Method

In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (Gʹ/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Burgers’ Equation have been discussed.


Introduction
Nonlinear partial differential equations have shown a variety of applications in almost every field, such as in electromagnetic, acoustics, electrochemistry, cosmology, biological and material science [1][2][3][4].Fractional differential equations can be considered as the generalization form of the differential equations, as they are involved with the derivatives of any real or complex order (for details see [3]).
In literature a lot of work has been done on nonlinear partial differential equations [5][6][7][8] rather than nonlinear partial differential equations (NPDEs) of fractional order.Recently, some new techniques have been introduced, by different authors, to find traveling wave solutions for NPDEs of fractional order [9][10][11].
In this article, the   G G  -expansion method [5] has been applied in the sense of modified Riemann-Liouville derivative to find the exact solutions of space-time fractional nonlinear Burgers' Equation, which has the following form [12]: The rest of the letter is organized as follows, in Section 2 the basic definitions and properties of the fractional calculus are considered regarding to modified Riemann-Liouville derivative.In Section 3, the   G G  -expansion method has been proposed to find the exact solutions for NPDEs of fractional order with the help of fractional complex transformation.As an application, the new exact solutions of nonlinear Burgers' Equation have been discussed in Section 4. In last Section 5, the conclusion has been drawn.

Preliminaries and Basic Definitions
In this section, the method has been applied in the sense of the Jumarie's modified Riemann-Liouville derivative [13,14] of order  .For this, some basic definitions and properties of the fractional calculus theory are being considered (for details see [3]).Thus, the fractional integral and derivatives can be defined following [13,14] Moreover, some properties for the modified Riemann-Liouville derivative have also been given as follows:

The (Gʹ/G)-Expansion Method for Nonlinear Fractional Partial Differential Equations
In this section, the   G G  -expansion method [15,16] has been discussed to obtain the solutions of nonlinear fractional partial differential equations.
For this, we consider the following NPDE of fractional order: where u is an unknown function and P is a polynomial of u and its partial fractional derivatives along with the involvement of higher order derivatives and nonlinear terms.
To find the exact solutions, the   G G  -expansion method can be performed using the following steps.
Step 1: First, we convert the NFPDE into nonlinear ordinary differential Equations using fractional complex transformation introduced by Li et al. [17].
The travelling wave variable where , K L and M are non-zero arbitrary constants, permits us to reduce Equation ( 2) to an ODE of If the possibility occurs, then Equation ( 4) can be integrated term by term one or more times.
Step 2: Suppose that the solution of Equation ( 4) can be expressed as a polynomial of   where i   s are constants and   G  satisfies the following second order linear ordinary differential equation.
with  and  as constants.
Step 3: The homogeneous balance can be used, to determine the positive integer m , between the highest or- der derivatives and the nonlinear terms appearing in (4).
Moreover, the degree of   u  can be defined as , which gives rise to the degree of the other expressions as follows: Therefore, the value of m can be obtained for the Equation (5).
Step 4: After the substitution of ( 5) into (4) and using Equation ( 6), we collect all the terms with the same or- Equate each coefficient of the obtained polynomial to zero, yields the set of algebraic equations for , , , , Step 5: After solving the system of algebraic equations, and using the Equation ( 6), the variety of exact solutions can be constructed.

Application of Burgers' Equation
In this section, the improved   G G  -expansion method have been used to construct the exact solutions for nonlinear space-time fractional Burgers' Equation (1).
It can be observed that the fractional complex transform where K and L are constants, permits to reduce the Equation ( 7) into an ODE.After integrating once, we have the following form: where C is a constant of integration.Now by considering the homogeneous balance between the highest order derivatives and nonlinear term presented in the above Equation, we have the following form where and L are arbitrary constants.To determine these constants substitute Equation (10) into (9), and collecting all the terms with the same power of   G G  together, equating each coefficient equal to zero, yields a set of algebraic Equations.
After solving these algebraic equations with the help of software Maple, yields the following results.Case 1 For the values: Straightforward simplification of Equation ( 10) yields, the following equation From Equations ( 6) and ( 11), we have the following travelling wave solutions.If 2 4 > 0    , then we have the following hyperbolic solution and if 2 4 0     , then we have the following solution Case 2 For the values: Straightforward simplification of Equation ( 10) yields, the following equation From Equations ( 6) and ( 12), we have the following travelling wave solutions.If 2 4 > 0    , then we have the following hyperbolic solution and if 2 4 0     , then we have the following solution where A and B are arbitrary constants.

Conclusion
The   G G  -expansion method has been extended to solve the nonlinear partial differential equation of fractional order, in the sense of modified Riemann-Liouville derivative.First, the fractional complex transformation has been used to convert the fractional equations into ordinary differential equation.Then   G G  -expansion method has been used to find exact solutions.As an application, the new exact solutions for the space-time fractional Burgers' Equations have been found.It can be concluded that this method is very simple, reliable and proposes a variety of exact solutions to NPDEs of fractional order. as: then we have the following trigonometric solution