_{1}

By introducing and extending the G'/G expansion method with the aid of computer algebraic system “Mathematics”, the exact general solutions were obtained for the Burgers-Huxley equation and special form. Final results were represented in hyperbolic function, trigonometric function and rational function with arbitrary parameters.

For the Burgers-Huxley equation

This an important equation used to describe the nonlinear diffusion phenomenon. In recent years, with the development of Symbolic Computation System and its perfection, people put forward a number of methods for solving the nonlinear equations of mathematical physics, such as the homogeneous balance method, F-method, Tanh method, projective Riccati method, ADM method [

This article will make the G'/G expansion method extended further, solving the Burger-Huxley equation [

Given nonlinear PDE, containing two independent variables x and t:

Among them, P is the polynomial of variable element u with high order partial derivative term and nonlinear term. For equations by G'/G expansion method (2) comprises the following steps:

1) On Equation (2) traveling wave reduction, let

Among them,

2) The

The

The positive integer n can be determined by the balance principle homogeneous.

3) We will plug Equation (5) into Equation (4), then the left of Equation (4) translates the polynomial of G'/G, making this polynomial coefficients are all zero, can obtain the algebraic equations about

4) With the help of Mathematica, we can solve the algebraic equations. so we can obtain the exact traveling wave solutions of Equation (1) that plugging the resulting value of

Make Burger-Huxley Equation for wave reduction, let

so Equation (1) can translate the ODE equation;

We can obtain a series of expansion of n is 1, assuming Equation (8) has the following form solution:

Among them,

In Equation (9) and Equation (10),

Plugging Equation (11) into Equation (8), Equation (8) can be transformed into a polynomial about G'/G expansion. Merger these items with respect to G'/G expansion which have the same power, and its coefficient is zero. We can obtain the equations as follows.

If set

1)

2)

, (13)

By using mathematics software to calculate again, a general solution of Equation (10) can be represented as:

Substituting Equation (13) and Equation (14) into Equation (9), traveling wave solution of Equation (1) can be obtained:

1) When

where

2) When

where

3) When

where

When the original equation of the parameters in the

Case 1: when

We assume that the same G'/G expression, by substitution to Equation (15) to obtain the following equations:

With the aid of Mathematica software, we can get the value of

1)

2)

3)

Equation (15) can be expressed as the solution:

1) When

where

2) When

where

3) When

where

Case 2: when

Make the equation to traveling wave reduced, let

Make Equation (19) to integral, so

We assume that the same G'/G expression, through to obtain the following equations

With the aid of mathematica software, obtain the solution of the following:

(exclude)

so the solution of Equation (18) can be expressed:

Including to Equation (15), we can obtain the accurate solution of Equation (18):

1) When

where

2) When

where

3) When

where

Based on the homogeneous balance method, the article obtains solutions of the Burgers-Huxley equation and two kinds of transformation type by the G'/G expansion method, making the Burgers-Huxley equation and its derivative equation solution in the form of more abundant. At the same time, we can obtain the hyperbolic traveling wave solutions of the equation and find the G'/G expansion method [

Mingxing Zhu, (2016) Solving the Burgers-Huxley Equation by G'/G Expansion Method. Journal of Applied Mathematics and Physics,04,1371-1377. doi: 10.4236/jamp.2016.47146