Advances in Mathematical Physics

Volume 2013 (2013), Article ID 890784, 11 pages

http://dx.doi.org/10.1155/2013/890784

Research Article

A New Method with a Different Auxiliary Equation to Obtain Solitary Wave Solutions for Nonlinear Partial Differential Equations

Bülent Kiliç and Hasan Bulut

Department of Mathematics, Firat University, 23119 Elazig, Turkey

Received 3 February 2013; Revised 22 April 2013; Accepted 23 April 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 Bülent Kiliç and Hasan Bulut. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A new method with a different auxiliary equation from the Riccati equation is used for constructing exact travelling wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of a different auxilliary equation from the Riccati equation which has more new solutions. More new solitary solutions are obtained for the RLW Burgers and Hirota Satsuma coupled equations.

1. Introduction

In the recent years, remarkable progress has been made in the construction of the exact solutions for nonlinear partial differential equations, which have been a basic concern for both mathematicians and physicists [1–3].We do not attempt to characterize the general form of nonlinear dispersive wave equations [4, 5]. When an original nonlinear equation is directly calculated, the solution will preserve the actual physical characters of solutions [6]. The studies in finding exact solutions to nonlinear differential equation (NPDE), when they exist, are very important for the understanding of most nonlinear physical phenomena. There are many studies which obtain explicit solutions for nonlinear differential equations. Many explicit exact methods have been introduced in literature [7–21]. Some of them are generalized Miura transformation, Darboux transformation, Cole-Hopf transformation, Hirota’s dependent variable transformation, the inverse scattering transform and the Bäcklund transformation, tanh method, sine-cosine method, Painleve method, homogeneous balance method (HB), similarity reduction method, improved tanh method and so on.

In this article, the first section presents the scope of the study as an introduction. In the second section contains analyze of a new method and balance term definition. In the third section, we will obtain wave solutions of RLW Burgers and Hirota Satsuma coupled equations by using a new method. In the last section, we implement the conclusion.

2. Method and Its Applications

Let us simply describe the method [22]. Consider a given partial differential equation in two variables

The fact that the solutions of many nonlinear equations can be expressed as a finite series of solutions of the auxiliary equation motivates us to seek for the solutions of (1) in the form

where,,   and are the wave number and the wave speed respectively, is a positive integer that can be determined by balancing the linear term of highest order with the nonlinear term in (1), is balancing coefficient that will be defined in a new “Balance term” definition and ,  ,   are parameters to be determined. Substituting (2) into (1) yields a set of algebraic equations for ,  ,   because all coefficients of have to vanish. From these relations ,  ,   can be determined. The main idea of our method is to take full advantage of the new auxiliary equation. The desired auxiliary equation presents as following

where and , , are constants.

Case 1. If , , then (3) has the solution .

Case 2. If , , then (3) has the solutions or .

Case 3. If , , then (3) has the solutions or

Remark 1. Depending on the and coefficients in the (3), it could be reached only three cases.

In the following we present a new approach to the “Balance term” definition.

Definition 2. When (1) is transformed with, , where and are real constants, we get a nonlinear ordinary differential equation for as following

Let is the highest order derivative linear term and is the highest nonlinear term in (5) and is the auxiliary equation that is used to solve the nonlinear partial differential equation then the “Balance term” can be decided by the balancing the nonlinear term and the linear term with acceptances of and where is integer and is the balance coefficient that can be determined later.

Example 1. For the KdV equation with the transform , we have the ordinary differential equation as following

By the balancing linear term with nonlinear term

we have the equations above and the equating to

If it is noticed that our new balance term is connected to . Namely our new balance term definition is connected to chosen auxiliary equation.

3. Application of the Method

Example 2. Let’s consider RLW Burgers equation

with the transform , we have the following equation

From the Definition 2 we have the balance term of RLW Burgers equation by using the auxiliary equation (3) for “Case 1”, is equal to 4. Therefore, we may choose the following ansatz:

Substituting (11) into (10) along with (3) and using Mathematica yields a system of equations w.r.t. . Setting the coefficients of in the obtained system of equations to zero, we can deduce the following set of algebraic polynomials with the respect unknowns , , for the Case 1:

From the system of (12) we have

(i)

(ii)

From the Definition 2 we have the balance term of RLW Burgers equation by using the auxiliary equation (3) for “Case 2”, is equal to −4 then we have the following system of equations

From the system of (15) we have

(i)

or

Example 3. Let’s consider Hirota Satsuma coupled equation

with the transform , , we have

From the Definition 2 we have the balance term of Hirota Satsuma coupled equation by using the auxiliary equation (3) for “Case 1”, is equal to 4 for and . Therefore, we may choose the following ansatz:

Substituting (20) into (19) along with (3) and using Mathematica yields a system of equations w.r.t. . Setting the coefficients of in the obtained system of equations to zero, we can deduce the following set of algebraic polynomials with the respect unknowns , , for the Case 1:

From the system of (21) we have

(i)

(ii)

From the Definition 2 we have the balance term of Hirota Satsuma coupled equation by using the auxiliary equation (3) for “Case 2”, is equal to −4 then we have the following system of equations

From the system of (3) we have

(i)

or

Figures 1, 2, 3 and 4 gives to us 3D graphics for RLW Burgers and RLW Burgers and Hirota Satsuma coupled equations.

(a)

(b)

Figure 1: Graph of the solution from left to the right for (13) and (14).

(a)

(b)

Figure 2: Graph of the solution from left to the right for (15) and (16).

(a)

(b)

(c)

(d)

Figure 3: Graph of the solutions and corresponding to the value from left to the right for (22), (3), and (23).

(a)

(b)

(c)

(d)

Figure 4: Graph of the solutions and corresponding to the value from left to the right for (25) and (26).

4. Conclusion

We have presented a new method and balance term definition and used it to solve the RLW Burgers and Hirota Satsuma coupled equations. In fact, this method is readily applicable to a large variety of nonlinear PDEs. First, all the nonlinear PDEs which can be solved by the other methods can be solved by our method. Second, we used only the special solutions of (3). If we use other solutions of (3), we can obtain more travelling wave solutions. Third it is a computerizable method, which allow us to perform complicated and tedious algebraic calculation on computer and so our balance term definition is effectively useful for any to chosen auxiliary equation.

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