1. 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[11-14], 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

(1)

where the dependent variable 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 in the form:

(2)

whereis the unknown function depending on space variable X and time variable t, and p is a polynomial in 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 so that , where is a constant. Based on this, we use the following changeof partial derivatives

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.Eq. (3) is then integrated as long as all terms contain derivatives, where integration constants are considered zeros.

Now, we suppose that the solution of the ODE (3) can be expressed by apolynomialin )as follows:

(4)

where satisfies the second order linear ODE in the form

(5)

where , and and are 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)

and for

(9)

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)into (3)with the value of m obtained in Step 1, and collecting all term(s) with the same order oftogether, the left-hand side of(3) converts into polynomial in . Then setting coefficients of to zero, we obtaina set of algebraic equations in.

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 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.

3.1 The B (n, 1) Burgers’ equation

This B(n, 1) Burgers’ equation is given as

(10)

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

(11)

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

(12)

Now balancingand , we obtain

A necessary condition for obtaining a closed form analytic solution is that m must a positive integer. Using the transformation

(13)

(12) converts to

(14)

Now, balancing and we obtain 2m=m+1 so.

Writing the solution of (14) in the form

(15)

Using (4) and (15), we obtain

(16)

(17)

Substituting (15), (16) and (17) into (14), and setting the coefficients of to zero, we obtain a system of algebraic equations in as follows:

(18)

(19)

(20)

Solving the system of equations (18)-(20) by Mathematica, we obtain

(21)

By using (21), the expression (15) can be written as

(22)

Where is defined by (8) and (9). Nowusing

(23)

When we obtain hyperbolic function solution of (10) as

(24)

When , we obtain trigonometric functionsolution of (10) as

(25)

where, and and are arbitrary constants and

If we set in (24), then soliton solution of (10) is obtained as

(26)

If we setin (24), then soliton solution of (10) is obtained as

(27)

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.

3.2.The B(-n,1) Burgers’ equation

The B(-n,1) Burgers’ equation is

(28)

Proceeding as earlier, we obtain theODE

(29)

Now balancing and , we obtain

Now using the transformation in (29), we obtain

(30)

Now balancing and we obtain 2m=m+1so. Setting the coefficients of to zero, we obtain a system of algebraic equations in as follows:

(31)

(32)

(33)

Solving the system of equations (31)-(33) by Mathematica, we obtain

Where free parameter, now using , the solution of (28) is given by

(34)

where is defined by (8) and (9).

When

(35)

When

(36)

where, and andare arbitrary constants and

If we set andthen (35) yields

(37)

Similarly, if we set and then (35) yields

(38)

We observe that the solutions (37) and (38) obtained for Burgers’ equation (28) are particular cases of solution (35). These solutions are exactly same as the solutions obtained by Wazwaz[18] for the above values of constants.

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.