FOROZANI Gh.and GHORVEEI NOSRAT M.

Department of Physics,Faculty of Science,Bu-Ali Sina University,Hamedan,Iran.

The Tanh Method for Kink Solution of Some Modified Nonlinear Equation

FOROZANI Gh.∗and GHORVEEI NOSRAT M.

Department of Physics,Faculty of Science,Bu-Ali Sina University,Hamedan,Iran.

Received 22 April 2013;Accepted 28 February 2014

.Thetanhmethod isaverypowerfultechnique forcomputation ofexacttraveling wave,in this paper this method has been employed for special modified states of Burger,Klein-Gordon and Fisher-Burger equations and the solitary solution of these equations are derived.

Tanh method;nonlinear equation;solitary solution.

1 Introduction

The nonlinear phenomena are very important in a variety of scientific fields,especially in fluid mechanics,solid state physics,plasma physics,plasma waves,nonlinear optics and etc[1].A variety of powerful method such as the inverse scattering[2],the Backlund transformation[3,4],sine-cos method[5],tanh-sech method[6],Hirota’s bilinear technique[7]and the homogeneous balance method[8]were used to solve nonlinear equations.The tanh method developed by Malfliet et al.[9,10],is a reliable and accurate algebraic method to obtain exact solution of nonlinear equations[11].Wazwaz also used this method for several forms of nonlinear partial differential equations such as

Fisher-Burger equation:

Klein-Gordon equation:

Burger equation:

Fisher-Burger equation has important applications in various fields such as traffic flow, financial mathematics,gasdynamic,appliedmathematics andphysics[12-19].Thisequationshowsaprototypicalmodelfordescribingtheinteractionbetweenthereactionmechanism,convection effect,and diffusion transport[20].In this paper solitary solution of theirs special modified states will be obtained.

2 Tanh method

Apartial differential equation(PDE)can beconvertedtoanordinarydifferential equation (ODE)upon using a wave variable[21]:

Introducing a new independentvariable y=tanhµz that leads to change of derivatives[7]:

Introduce the ansatz:

where m in most cases is a positive integer.To determine the parameter m we usually balance the linear terms of highest order in the resulting equation with the highest order nonlinear terms.With m determined,equate the coefficients of powerof y in the resulting equation[1].

3 Modified nonlinear Burgers equation

By adding a nonlinear term of the form u(1-u2)to burger’s equation,the modified burger equation is obtained as follows:

Substituting(2.1)into(3.1)gives:

The m value is determined by balancing u3with u′′it is easily to shown that m is equal to 1.So that:

After substituting(3.3)into(3.2)and collecting the coefficients of each power of y,we find the system of algebraic equations for a0,a1,c andµ:

By using a simple computational program,we find:

so that:

The plot of Eq.(3.5)is shown in Fig.1 and as can be seen it represent a kink.

4 Modified nonlinear Klein-Gordon equation

We now consider the modified nonlinear Klein-Gordon equation by adding a term in form uxxxto nonlinear Klein-Gordon equation:

By using wave variable(2.1)we have:

Figure 1:The plot of Eq.(3.5).

Balancing u′′with u4gives m=1 so that:

Proceeding as before we obtain the system of algebraic equations for a0,a1,µand α:

We find:

so that:

The diagram of Eq.(4.5)is shown in Fig.2 for c=2,a,α and γ=1.This diagram shown an antikink.

Figure 2:The plot of Eq.(4.5).

5 Modified nonlinear Fisher-Burgers equation

The modified Fisher-burger equation can be obtained by adding also a nonlinear term of the form αu2(1-u2)as follows:

Substituting(2.1)into(5.1)gives ODE:

Balancing u′′whit u4gives m=1 so that:

Proceeding as before we obtain the system of algebraic equations for a0,a1andµ:With value ofwe find:

We find:

The diagram of Eq.(5.9)is shown in Fig.3.This diagram shown an antikink.

Figure 3:The plot of Eq.(5.9).

6 Discussion

The obtained results clearly demonstrate that the tanh technique is a powerful solution method to find analytical expressions for variety nonlinear equations.

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10.4208/jpde.v27.n2.4 June 2014

∗Correspondingauthor.Emailaddresses:g.forozani@gmail.com(Gh.Forozani),m.ghorveei@yahoo.com(M. Ghorveei Nosrat)

AMS Subject Classifications:76B25,37K40,34A34,34G20

Chinese Library Classifications:O175.14