The (G`/G)-Expansion Method for Traveling Wave Solutions of Burgers’ Kdv and Generalization of Huxley Equations

The (G`/G)-expansion method is used for determining the exact traveling wave solutions of the Burgers-KdV and generalization of Huxley equations. The obtained solutions are compared with the solutions found by Wazwaz[18]. The (G`/G)-method is very powerful and easy tool for solving non-linear partial differential equations


Introduction
Nonlinear equations in mathematical physics appear in various areas, such as fluid dynamics, plasma physics, optical fibers, solid state physics and other applications. A variety of powerful methods have been used to study the nonlinear equations such as the homotopy perturbation method [1], the variational iteration method [2], Hirotas bilinear methods [3], the sine-cosine function method [4][5], Jacobi elliptic method [6], the standard tanh and extended tanh methods [7][8][9][10][11], the exp-function method [12][13], the inverse scattering method [14] and so on. One of the most powerful and direct methods for constructing solutions of non-linear equations is the (G`/G)-expansion method [15][16][17]. This method was first introduced by Wang et al. [15] and it has been widely used for finding various exact solutions of non-linear partial differential equations. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations The parameter m plays an important role in the (G`/G)expansion method; it should be a positive integer to derive a closed form analytic solution. However, for non-integer values of m , we usually apply a transformation formula to overcome this difficulty.
We begin our study by examining the Burgers-KdV and generalization of Huxley equation [18] ( ) 0, 1, , 0 u au u k u u n a

Analysis of the (G`/G)-expansion method
The (G`/G)-expansion method ( [15][16][17]) is a powerful solution method for the computation of exact traveling wave solutions of partial differential equations (PDEs).
We consider non-linear PDE for ) , ( t x u in the form: ( , , , , , ,...) 0, is the unknown function depending on space variable x and time variable , t P is a polynomial in ) , ( t x u and its partial derivatives, in which the highest order derivatives and nonlinear terms are involved. To obtain the traveling wave solution of PDE (2), we introduce the variable ω is a constant. We use the following change of partial and so on for the other derivatives. Thus PDE (2) reduces to an ordinary differential equation (ODE) ' '' ''' ( , , , ,...) 0, Q U U U U = (3) where the primes denote the derivative with respect to ξ . Equation (3) is then integrated as long as all the terms contain derivatives, where integration constants are considered to be zero. Now, we assume that the solution of the ODE (3) can be expressed by a polynomial in (G`/G) as follows: Using the general solution of (5), we have for 2 Also for , 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 2. By substituting (4) and (5)  Step 4. By substituting the results obtained in the above steps, we can obtain exact traveling wave solutions of (2).

Applications
In this section, we apply the (G`/G)-expansion method to construct the traveling wave solution of Burgers-KdV and generalization of Huxley equations.

The Burgers-Kdv equation
The Burgers-Kdv equation is given by where the primes denote the derivative with respect to .

Generalization of Huxley equation
Consider a generalization of the Huxley equation .

Conclusions
(G`/G)-expansion method is used to obtain the exact solutions of the Burgers-KdV and generalization of Huxley equations. The solution method is very simple and effective. 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. The solutions are compared with the solutions obtained by Wazwaz [18], and it is found that the solutions obtained are exactly same as determined by Wazwaz [18].