General Solution of Two Generalized Form of Burgers Equation by Using the G G-Expansion Method

In this work, the G G        -expansion method is proposed for constructing more general exact solutions of two general form of Burgers type equation arising in fluid mechanics namely, Burgers-Korteweg-de Vries (Burgers-KdV) and Burger-Fisher equations. Our work is motivated by the fact that the G G        -expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.


Introduction
Nonlinear evolution equations (NLEEs) have been the subject of study in various branches of mathematicalphysical sciences such as physics, biology, chemistry, etc.The analytical solutions of such equations are of fundamental importance since a lot of mathematical-physical models are described by NLEEs.Among the possible solutions to NLEEs, certain special form solutions may depend only on a single combination of variables such as traveling wave variables.In the literature, there is a wide variety of approaches to nonlinear problems for constructing traveling wave solutions.Some of these approaches are the Jacobi elliptic function method [1], inverse scattering method [2], Hirotas bilinear method [3], homogeneous balance method [4], homotopy perturbation method [5], Weierstrass function method [6], symmetry method [7], Adomian decomposition method [8], sine/cosine method [9], tanh/coth method [10], the Exp-function method [11][12][13][14][15][16] and so on.But, most of the methods may sometimes fail or can only lead to a kind of special solution and the solution procedures become very com-plex as the degree of nonlinearity increases.

  
The degree of the polynomial can be determined by considering the homogeneous balance between the highest order derivative and nonlinear terms appearing in the given nonlinear evolution equations.The coefficients of the polynomial can be obtained by solving a set of algebraic equations resulted in from the process of using the method.The G   -expansion method is direct, concise, G     elementary and effective, and can be used for nonlinear , much work has been done on developing an partial differential equation involving higher-order nonlinear terms.
As we know d extending the G   -expansion method for solving G     nonlinear evolution equations (see, for example [17][18][19][20][21][22][23][24][25][26][27] (2) and their references).But in generalized cases, a small amount of work has been done (see, for example [28,29]).In this paper, we restrict our attention to the study of the following generalized forms of Burgers equation, where and and are constants.These ns are t al m Equation ( 4) is the lowest order approx on > 1, n , p q r equatio he gener ized co bined forms of Burgers equation = 0 imation for the e-dimensional propagation of weak shock waves in a fluid [30,31].It is also used in the description of the variation in vehicle density in highway traffic [32].It is one of the fundamental model equations in fluid mechanics.The Burgers equation demonstrates the coupling between diffusion xx u and the convection process x uu .Burgers introduce his equation to capture some of features of turbulent fluid in a channel caused by the interaction of the opposite effects of convection and diffusion [33].It is also used to describe the structure of shock waves, traffic flow, and acoustic transmission [34].
The generalized Burgers-KdV Equation ( 2) are models d t the for the propagation of waves on an elastic tube (see [28,29,35,36] and their references).It can be regarded as a combination of the Burgers equation ( 0 -exp thod for the Burgers-KdV equation ( and in [36] tion terest in the present work is in implementin ansion me-= 1 n ) to m Ahmet Bekir applied the tanh method odified form of Burgers-KdV equation ( = 2 n ).In addition, the generalized Burgers-Fisher Equa (3), see [37], has a wide range of applications in plasma physics, fluid physics, capillary-gravity waves, nonlinear optics and chemical physics [38].

Our first in g the G G
   -expansion method to stress its power in     handling nonlinear equations, so that one can apply it to models of various types of nonlinearity.The next interest is in the determination of exact travelling wave solutions for the generalized form of Equations ( 2 where U U  and prime denotes derivative with respect sume that the solution of Equation ( 6) can be expressed by a polynomial in to ξ.We as G     as follows: where , and i  0  are constants to be determined later, and   G  satisfies the second order linear ordinary di- ffere quation (1 where and   are arbitrary constants.It follows ) a th from (7 nd (8), at and so on, here the prime denotes the derivative with respective to  .
To determine explicitly, we take the following er ne the integer by substituting Equat a ar term(s) and the highest order derivative.al u three steps: Step 1. Det mi m ion (7) into Equ tion (6) and balancing the highest order nonline Step 2. Substitute Equation ( 7) ong with Equation ( 8) into Equation ( 6) and collect all terms with the same . Then set each coefficient of this polynomial to zero to derive a set of algebraic equations for 0 , , , ,

Application
In this section, we will -expansion method on the generalized equations listed in (2) the generalized Burgers-KdV equation with and (3).

Generalized Burgers-KdV Equation
Considering higher-order nonlinear terms = 0 Now,using the wave variable     , = , ation and ne in (11) and in glecting the constant find tegrating the of integration, we resulting equ-


To achieve our goal, we use the transformation According to Step 1, we get We then suppose that E has the following formal solutions: where 2 Substitute the above general case in (10), we get then use the transformation bolic function solutions of Equation ( 11), becomes: that will carry (12) and the trigonometric function solutions of Equation (11), will be: where and when then we deduce from general solutions (17)-( 18) that, respectively (20) where are the four solitary wave solutions of the particular Burgers-KdV equation According to Step 1, we get 2, hence We then suppose that following formal solutions: is arbitrary constant and nt real solutions of th urgers-KdV Equation ( 2) were obtained by Ahmet Bekir in [36] using the extended tanh method.

Generalized Burgers-Fisher Equation
Now considering the generalized Burgers-Fisher equation with higher-order nonlinear terms 0 (24 where and 0  ed later.
  hieve nsfor To ac our goal, we use the tra mation are constants hich are un-known to be determin Similar on previous section, substituting Equation (29) into Equation ( 26) and collecting all terms with the same , together, we derive a set of algebraic equations for Substitute the above general case in (25), we get then use the transformation  bolic and rigonometric function solutions of Equa-the t tion (11), will be:   and when then we deduce from general solutions ( 30)-( 31) that, respectively ( 1) where are the four solitary wave solutions of the particular Burgers-Fisher equation is arbitrary constant and Different real solutions o neral Burgers-Fisher Equation (3) were obtained by Wazwaz in [39] using the tanh method and by El-Wakil in [38] using a modified tanh-function method and recently by Luwai Wazzan in [28] using a modified tanh-coth method.

Conclusions
This study shows that the -expansion method is quite efficient and practically well suited for use in finding exact solutions for the generalized form of Burgers-Korteweg-de Vries (Burgers-KdV) and Burger-Fisher equations.The results show that this method is a powerful Mathematical tool for obtaining exact solutions for the general B-Fisher and B-KdV equations.With the aid of Maple, we have assured the correctness of the obtained solutions by putting them back into the original equation.
It is also a promising method to solve other nonlinear partial differential equations arising in engineering sci-

Acknowledgements
This work is partially supported by Grant-in-Aid fro the rdabil, Iran.
more travelling wave solutions of Equation(5).

 and 1 ,
and solving them by use of Maple, we get the following general result: