Extended Improved tanh-Function Method for Solving the Nonlinear Physical Problems
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Extended Improved tanh-Function Method for Solving the Nonlinear Physical Problems A.A. Soliman
Received: 23 May 2008 / Accepted: 18 June 2008 / Published online: 10 July 2008 © Springer Science+Business Media B.V. 2008
Abstract By means of the extended improved tanh-function (EITF) method with computerized symbolic computations for constructing new multiple traveling wave solutions for some different kinds of nonlinear physical problems are presented and implemented. The solutions for the nonlinear equations such as Coupled MKdV and Coupled Hirota-Satsuma Coupled KdV equations which include new trigonometric function solutions and rational solutions are exactly obtained. So consequently, the efficiency of the method can be demonstrated. Keywords Coupled MKdV equations · Hirota-Satsuma Coupled KdV equations · EITF method
1 Introduction The nonlinear phenomena are very important in a variety of scientific fields, especially in fluid dynamics, solid state physics, hydrodynamic, plasma physics, nonlinear optics, etc. Analytical solutions to nonlinear partial differential equations play an important role in nonlinear science, especially in nonlinear physical science since they can provide much physical information and more inside into the physical aspects of the problem and thus lead to further applications. Much work has been down over the years on the subject of obtaining the analytical solutions to the PDEs. Many powerful methods to seek for exact solutions to the nonlinear differential equations have been proposed. Among these are Backlund transformation [1–4], the tanh method [5], the sine-cosine method [6, 7], the homogeneous balance method [8, 9], the Riccati expansion method with constant coefficients [10, 11], variational iteration methods [12–15], collocation method [16–18] and sine-Gordon equation expansion method [19, 20]. A.A. Soliman () Department of Mathematics, Faculty of Education (AL-Arish), Suez Canal University, AL-Arish, 45111, Egypt e-mail: [email protected]
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A.A. Soliman
Recently the modified extended tanh-function method and symbolic computation have been suggested for solving the system of nonlinear partial differential equations [21–23] and nonlinear equations of special interest in physics namely, the Broer-KaupKupershmidt,nonlinear coupled plasma, and coupled-nonlinear reaction-diffusion equations [24]. The aim of this paper is extending of the improved tanh-function method to solve two different types of nonlinear physical equations such as Coupled MKdV and Hirota-Satsuma Coupled KdV equations.
2 Extended Improved tanh-Function Method To illustrate the basic concepts of the EITF method. We consider a given PDE in two independent variables given by F (u, ux , ut , uxx , . . .) = 0.
(1)
We first consider its travelling solutions u(x, t) = u(ξ ), ξ = x + ct or ξ = x − ct , then (1) becomes an ordinary differential equation. In order to seek the travelling wave solutions of (1), we introduce the following u(ξ ) = a0 +
M
ai φ i +
i=1
M
bi φ −i ,
(2)
i=1
φ = α + βφ +
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