1. 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[7-11], 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- 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 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 , we usually apply a transformation formula to overcome this difficulty.

(1)

2 Analysis of the (G`/G)-expansion method

(2)

and so on for the other derivatives. Thus PDE (2) reduces to an ordinary differential equation (ODE)

(3)

where the primes denote the derivative with respect to. 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:

(4)

where satisfies the second order linear ODE in the form

(5)

where , and ,

andare real constants which are to be determined.

Using (4) and (5), we obtain

(6)

(7)

Using the general solution of (5), we have for

(8)

for

(9)

Also for

To determine U explicitly, we take the following four steps:

Step 1. Determine the integer by substituting (4) along with (5) into (3), and balancing the highest order nonlinear term(s) and the highest order partial derivative.

Step 3. Solve the system of algebraic equations obtained in step 2 for and by use of Mathematica.

Step 4. By substituting the results obtained in the above steps, we can obtain exact traveling wave solutions of (2).

3. 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.

3.1. The Burgers-Kdv equation

The Burgers-Kdv equation is given by

(10)

Using the transformation, where, the PDE is reduced to an ODE

(11)

where the primes denote the derivative with respect to. Integrating once with respect to and taking constant of integration to be zero, (11) reduces to

(12)

Balancing and,

(13)

Using the transformation

(14)

equation (12) converts to

(15)

Now balancing with i.e. , we obtain =1. Therefore, we assume the solution of (15) in the form

(16)

Using (6), (7) and (16), we obtain

(17)

(18)

(19)

(20)

Now using (16)-(20) in (15) and equating the coefficients of to zero, we obtain a system of algebraic equations in and as follows:

(21)

(22)

(23)

(24)

(25)

Solving the system of equations (21)-(25) by using MATHEMATICA, we obtain the following two sets of solutions for

Set1:

Set2:

Hence, for Set 1:

(26)

Now, using

(27)

When we obtain hyperbolic function solution of the Burgers-KdV equation (10) as

(28)

when 0, we obtain the trigonometric function solution of the Burgers-KdV equation (10) as

(29)

and when

(30)

If we set in (28), we obtain

(31)

If we set in (28), we obtain

(32)

(33)

for and

(34)

for

3.2. Generalization of Huxley equation

Consider a generalization of the Huxley equation

(35)

Proceeding as earlier, equation (35) is converted to the ODE

(36)

Now, balancing and we find

Using the transformation in (36), we obtain

(37)

Now again balancing with i.e. we obtain =1. Therefore, assumes the same form as in (16). Now, putting the different values of etc. from (16)-(20) in (37) and setting the coefficients of to zero we obtain a system of algebraic equations in and as follows:

(38)

(39)

(40)

(41)

(42)

Fixing and then solving the system of equations (38)-(42) by MATHEMATICA, we obtain two sets of solutions:

Set 1:

Set 2:

Therefore, the solution of the generalization of the Huxley equation (35) corresponding to Set 1 using

(43)

When

(44)

when

(45)

and when

(46)

If we set in (44), we obtain

(47)

If we set in (44), we obtain

(48)

Similarly for Set 2

(49)

for and

(50)

for

4. 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].