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ISRN Applied Mathematics
Volume 2012 (2012), Article ID 565247, 4 pages
http://dx.doi.org/10.5402/2012/565247
Research Article

Traveling-Wave Solution of Modified Liouville Equation by Means of Modified Simple Equation Method

Department of Computer Science and Engineering, Prime University, Dhaka-1216, Bangladesh

Received 2 August 2012; Accepted 21 August 2012

Academic Editors: J. Kou and S. Sture

Copyright © 2012 Md. Abdus Salam. 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

We construct the traveling wave solutions involving parameters of modified Liouville equation by using a new approach, namely the modified simple equation method. The proposed method is direct, concise, and elementary and can be used for many other nonlinear evolution equations.

1. Introduction

The investigation of the traveling-wave solutions of nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena. Several direct methods for finding the explicit traveling-wave solutions to nonlinear partial differential equations have been proposed, such as the tanh-function method and its various extension [1], the Jacobi elliptic function expansion method [2], the homogeneous balance method [3, 4], the F-expansion method and its extension [5], the variational iteration method [6], (𝐺/𝐺)-expansion method [7], and so on. More recently, a new method, named modified simple equation method [8, 9], has been proposed to construct more explicit traveling-wave solutions of modified Liouville equation.

2. Description of the Modified Simple Equation Method

Suppose that a nonlinear equation, say in two independent variables 𝑥 and 𝑡 is given by 𝑃𝑢,𝑢𝑡,𝑢𝑥,𝑢𝑡𝑡,𝑢𝑥𝑡,𝑢𝑥𝑥,=0,(2.1) where 𝑢=𝑢(𝑥,𝑡) is an unknown function, 𝑃 is a polynomial in 𝑢=𝑢(𝑥,𝑡) and its various partial derivatives, in which the highest-order derivatives and nonlinear terms are involved. In the following, the main steps of the modified simple equation method are given.

Step 1. The traveling-wave variable 𝑢(𝑥,𝑡)=𝑢(𝜉),where𝜉=𝐴𝑥+𝐵𝑡(2.2) permits us reducing (2.1) to an ODE for 𝑢=𝑢(𝜉) in the form 𝑃𝑢,𝑉𝑢,𝑢,𝑉2𝑢,𝑉𝑢,𝑢,=0.(2.3)

Step 2. Suppose that the solution of ODE (2.1) can be expressed by a polynomial in (𝜓/𝜓) as follows: 𝑢(𝜉)=𝑛𝑖=0𝛼𝑖𝜓𝜓𝑖,(2.4) where 𝛼𝑖 are arbitrary constants to be determined such that 𝛼𝑛0, while 𝜓(𝜉) is an unknown function to be determined later.

Step 3. We determine the positive integer 𝑛 by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in ODE (2.3).

Step 4. We substitute (2.4) into (2.3), we calculate all the necessary derivatives 𝑢,𝑢,, and then we account the function 𝜓(𝜉). As a result of this substitution, we get a polynomial of 𝜓/𝜓 and its derivatives. In this polynomial, we equate with zero all the coefficients of it. This operation yields a system of equations which can be solved to find 𝛼𝑖 and 𝜓(𝜉). Consequently, we can get the exact solution of (2.1).

3. Application of the Method

In this section, we would like to use our method to obtain new and more general exact traveling wave solutions of the modified Liouville equation 𝑤𝑡𝑡=𝑎2𝑤𝑥𝑥+𝑏𝑒𝛽𝑤,(3.1) where 𝑎,𝑏, and 𝛽 are arbitrary constants.

Suppose 𝑒𝛽𝑤=𝑢(𝑥,𝑡), where the traveling-wave transformation is 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝐴𝑥+𝐵𝑡.(3.2) By using the traveling-wave variable (3.2), (3.1) is converted into an ODE for 𝑢=𝑢(𝜉)𝑢𝑢𝑢2+𝑘𝑢3=0,where𝑘=𝑏𝛽𝑎2𝐴2𝐵2.(3.3) Suppose that the solution of the ODE (3.3) can be expressed by a polynomial in (𝜓/𝜓) as follows: 𝑢(𝜉)=𝑛𝑖=0𝛼𝑖𝜓𝜓𝑖,(3.4) where 𝛼𝑖 are arbitrary constants provided 𝛼𝑛0.

Considering the homogeneous balance between the highest order derivatives and the nonlinear terms in (3.3), we get 𝑛=2 and hence the solution takes the following form: 𝑢(𝜉)=𝛼0+𝛼1𝜓𝜓+𝛼2𝜓𝜓2,(3.5) where 𝛼20. On substituting (3.5) into the ODE (3.3) and equating all the coefficients of 𝜓1,𝜓2,𝜓3,𝜓4,𝜓5,𝜓6 to zero, we, respectively, obtain 𝑘𝛼30=0,(3.6)3𝑘𝛼20𝛼1𝜓+𝛼0𝛼1𝜓=0,(3.7)2𝛼0𝛼2𝜓2𝛼21𝜓23𝛼0𝛼1𝜓𝜓+3𝑘𝛼20𝛼2𝜓2+3𝑘𝛼0𝛼21𝜓2+2𝛼0𝛼2𝜓𝜓+𝛼21𝜓𝜓=0,(3.8)𝑘𝛼31𝜓3+2𝛼0𝛼1𝜓32𝛼1𝛼2𝜓𝜓2𝛼21𝜓2𝜓+3𝛼1𝛼2𝜓2𝜓10𝛼0𝛼2𝜓2𝜓+6𝑘𝛼0𝛼1𝛼2𝜓3=0,(3.9)2𝛼22𝜓2𝜓2+2𝛼22𝜓3𝜓+𝛼21𝜓4+3𝑘𝛼0𝛼22𝜓4+3𝑘𝛼21𝛼2𝜓4+6𝛼0𝛼2𝜓45𝛼1𝛼2𝜓3𝜓=0,(3.10)3𝑘𝛼1𝛼22𝜓52𝛼22𝜓4𝜓+4𝛼1𝛼2𝜓5=0,(3.11)𝑘𝛼32𝜓6+2𝛼22𝜓6=0.(3.12) Equations (3.6), (3.8), and (3.12) give 𝛼0=0,𝛼1=0, 𝛼2=2/𝑘(𝜓0, otherwise it is the trivial case); those satisfy (3.7) and (3.9), and (3.10), (3.11), respectively, yields 𝜓2𝜓𝜓𝜓=0,(3.13)=0.(3.14) Equation (3.14) gives 𝜓=0. Integrating 𝜓=0 with respect to 𝜉, we get 𝜓=𝐶1+𝐶2𝜉 and the solution of ODE (3.3) takes the following form: 𝑢(𝜉)=2𝑘𝐶1𝐶1+𝐶2𝜉2=2𝐵2𝑎2𝐴2𝐶𝑏𝛽𝐴𝑥+𝐵𝑡+𝐶2𝐶,where𝐶=1𝐶2.(3.15) And finally the traveling-wave solution of (3.1) is 1𝑤=𝛽2𝐵ln2𝑎2𝐴2𝐶𝑏𝛽𝐴𝑥+𝐵𝑡+𝐶2.(3.16)

4. Conclusion

On comparing this method with the other methods via the tanh-function method, homogeneous balance method, and the (𝐺/𝐺)-expansion method used in [1, 4, 7], we see that the modified simple equation method is much more simpler than these methods because these methods have used the computer programs, while the modified simple equation method has not used these programs. Also we deduce that the modified simple equation method is effective and standard which allows us to solve complicated nonlinear evolution equations in the mathematical physics.

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