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The novel (G'/G)-expansion method is a powerful and simple technique for finding exact traveling wave solutions to nonlinear evolution equations (NLEEs). In this article, we study explicit exact traveling wave solutions for the (1 + 1)-dimensional combined KdV-mKdV equation by using the novel (G'/G)-expansion method. Consequently, various traveling wave solutions patterns including solitary wave solutions, periodic solutions, and kinks are detected and exhibited.

NLEEs arise in a wide variety of disciplines physical problems such as in physics, biology, fluid mechanics, solid-state physics, biophysics, solid mechanics, condensed matter physics, plasma physics, quantum mechanics, optical fibers, elastic media, reaction-diffusion models, and quantum field theory. Recently, many kinds of powerful methods have been proposed to find exact traveling wave solutions of NLEEs e.g., the

For a given nonlinear wave equation with one physical field

where

Let us consider that the traveling wave variable is

The traveling wave variable (2), transforms (1) into a nonlinear ODE for

We seek for the solution of Equation (3) in the following generalized ansatze

where

Herein _{N} might be zero, but both of them could not be zero simultaneously. α_{j}

where prime denotes the derivative with respect to

The Cole-Hopf transformation

Equation (7) has individual twenty five solutions (see Zhu, [

The value of the positive integer N can be determined by balancing the highest order linear terms with the nonlinear terms of the highest order come out in Equation (3). If the degree of

Substituting Equation (4) including Equations (5) and (6) into Equation (3), we obtain polynomials in

sulted polynomials to zero, yields an over-determined set of algebraic equations for

Suppose the value of the constants can be obtained by solving the algebraic equations obtained in Step 4. Substituting the values of the constants together with the solutions of Equation (6), we will obtain some new and comprehensive exact traveling wave solutions to the nonlinear evolution Equation (1).

In this section, we will employ the novel (G'/G)-expansion method to get several novel and further wide-ranging exact traveling wave solutions to the famous (1 + 1)-dimensional combined KdV-mKdV equation.

Let us consider the (1 + 1)-dimensional combined KdV-mKdV equation

Using the traveling wave transformation

Integrating Equation (9), we obtain

where C is a constant of integration. Inserting Equation (4) into Equation (10) and balancing the highest order derivative

Therefore, the solution of Equation (10) takes the form,

Substituting Equation (11) into Equation (10), the left hand side is transformed into polynomials of

nomials to zero, we obtain an over-determine set of algebraic equations (for simplicity we leave out to display the equations) for

where d,

For Set, substituting Equation (12) and the values of

When

where

where A and B are real constants.

When

where A and B are arbitrary constants such that

When

where k is an arbitrary constant.

When

where

In this letter, the novel (G'/G)-expansion method has been successfully applied to find the exact solution for the (1 + 1)-dimensional combined KdV-mKdV equation. The novel (G'/G)-expansion method is used to find a new exact traveling wave solution. The results show that the novel (G'/G)-expansion method is reliable and effective tool to solve the (1 + 1)-dimensional combined KdV-mKdV equation. Thus the novel (G'/G)-expansion method could be a powerful mathematical tool for solving NLEEs.

The Authors offer sincere thanks to the referees and editorial board of JAMP for their helpful and kind support. Furthermore, Fethi Bin Muhammad Belgacem is pleased to acknowledge the continued support of the Public Authority for Applied Education and Training (PAAET RD) for their continued support.

Md. NurAlam,Fethi Bin MuhammadBelgacem,M. AliAkbar, (2015) Analytical Treatment of the Evolutionary (1 + 1)-Dimensional Combined KdV-mKdV Equation via the Novel (G'/G)-Expansion Method. Journal of Applied Mathematics and Physics,03,1571-1579. doi: 10.4236/jamp.2015.312181