Application of The (GG`/GG)-Expansion Method For Solving The Generalized Forms B(n,1) andB(-n,1) of Burgers’ Equation

In this paper, the exact traveling wave solutions of thegeneralized forms B(n, 1) and B(-n, 1) of Burgers’ equation are obtained by using (G`/G)-expansion method. It has been shown that the (G`/G)-expansion method, with the help of computation, provides a very effective and powerful tool for solving non-linear partial differential equations


Introduction
The study of non-linear partial differential equations plays a very important role in many fields of mathematics and physics such as fluid dynamics, plasma physics, astro physics, solid state physics and quantum field theory. To obtain the solutions of non-linear partial differential equations, many methods were used, such as the Similarity transformation method [6,10], Backlund transformation method [7], the sine-cosine method [3,4], the Jacobi elliptic function method [8], the tanh method [11][12][13][14], the exp-function method [9,17], the inverse scattering method [1], and so on. One of the most powerful and direct methods for constructing solutions of non-linear partial differential equations is the (G`/G)-expansion method [2,5,16]. This method is first introduced by Wang et al. [16], and it has been widely used for finding exact solutions of non-linear partial differential equations. In this method, the linearization of non-linear differential equations for traveling wave is performed with a certain substitution which leads to a.second-order differential equation with constant coefficients. The calculations are performed with a computer algebra system for finding solutions of the non-linear equations The generalized forms B(n,1) and B(-n,1) of Burgers' equation [18] are ( ) 0, 1, , 0, where the dependent variable u is a function of space variable x and time variable t, a and b are arbitrary constants.
The generalized forms of Burgers' equation appear in various areas of mathematics, such as in the modeling of fluid dynamics, the propagation of waves, and traffic flow. Here, our goal is to obtain the exact traveling wave solutions for the generalized forms of Burgers' equation by using (G`/G)-expansion method.

2.The (G`/G)-expansion method
The (G`/G)-expansion method ( [2,5,16]) is a powerful solution method for the computation of exact traveling wave solutions of partial differential equations (PDEs).
We consider the non-linear PDE for ) , ( t x u in the form: ( , , , , , ,...) 0, is the unknown function depending on space variable x and time variable t , and P is a polynomial in ) , ( t x u and its partial derivatives, in which the highest orderderivatives and nonlinear terms are involved. To find the traveling wave solution of PDE (2), we introduce the variable where ω is a constant. Based on this, we use the following changeof partial derivatives Using (4) and (5), we obtain Using the general solution of (5), we have for 2 and for 2 4 To determine U explicitly, we take the following four steps: Step 1. Determine the integer m by substituting (4) along with (5)into (3), and balancing the highest order nonlinear term(s) and the highest order partial derivative.
Step 4. By substituting the results obtained in the above steps, we can obtain a series of fundamental solutions of (3).

3.Applications
In this section, we apply the (G`/G)-expansion method to construct the traveling wave solution of generalized forms of Burgers' equation.

The B (n, 1) Burgers' equation
This B(n, 1) Burgers' equation is given as ( ) 0, n t x xx u a u bu + + = . 0 , , 1 ≠ > b a n (10) Using the transformation u(x, t)=U ( ) ξ ,where where primes denote the derivative with respect to ξ .Integrating once with respect to ξ and taking constant of integration to be zero, (11) Now balancing n U and ' U , we obtain 1 , 1 1 m n n = > − A necessary condition for obtaining a closed form analytic solution is that m must a positive integer. Using the transformation 1 1 n Now, balancing 2 V and ' V we obtain 2m=m+1 so 1 m = . Writing the solution of (14) in the form Using (4) and (15), we obtain Substituting (15), (16) and (17) b n b G V a n a n G λ ω ω ξ is defined by (8) and (9). Nowusing n b a n a n n b a n a n where, If we set 1 2 0, 0, (24), then soliton solution of (10) is obtained as If we set 1 2 C 0,C 0, = ≠ 0, n 2, b 1 λ = = = in (24), then soliton solution of (10) is obtained as We see that (26) and (27) are the particular cases of the general exact solution (24). The solutions (26) and (27) are exactly same solutions as obtained by Wazwaz [18] for the above values of constants. Hence, our solutions (24) and (25) are more general.

4.Conclusions
In this paper, we used the (G`/G)-expansion method to obtain the exact traveling wave solutions of the generalized forms of Burgers' equation. The solution method is very simple and effective, and the solutions are expressed in the form of hyperbolic functions and the trigonometric functions. It is shown that this method is a good tool for handling non-linear partial differential equations. Correctness of the solutions is also checked by comparing with the solutions obtained by Wazwaz [18], and substituting them back into the original equations with the help of Mathematica.