%0 Journal Article
%T The Extended Tanh Method and the Exp-Function Method to Solve a Kind of Nonlinear Heat Equation
%A Weimin Zhang
%J Mathematical Problems in Engineering
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/935873
%X We study a kind of nonlinear heat equation with temperature-dependent thermal properties by the aid of the extended Tanh method and the Exp-function method. We obtain abundant new exact solutions of the equation. By comparing both of the methods, we find that the Exp-function method gives more solutions in this problem. 1. Introduction The classical heat equation also known as the diffusion equation, describes in typical applications of the evolution in time of the density of some quantities such as heat and chemical concentration [1,page 44]. In this case, the thermal diffusivity and thermal conductivity of the medium are assumed to be constant. However, in some media such as gases, the parameters are proportional to the temperature of the medium giving rise to a nonlinear heat equation of the following form [2]: where is the conductivity, is diffusivity, and is a constant. When the diffusivity is proportional to , a more general nonlinear heat equation reads as In a recent paper [3], using the Adomian decomposition method, the author discussed the following nonlinear heat equation with temperature dependent diffusivity: where and . In this paper we are interested in the following nonlinear heat equation: and discuss its traveling wave solutions. As we know, a solution of the form is called a traveling wave (with wavefront normal to , velocity , and profile ) [1, page 172]. Here we employ, for the first time, the extended Tanh method and Exp-function method for solving (1.5), and abundant new exact solutions of (1.5) are presented. We compare both of the methods and find that the Exp-function method is more efficient than the extended Tanh method in this problem. 2. The Extended Tanh Method We now describe the extended Tanh method for the given partial differential equations. The Tanh method was defined by Malfliet [4] and Fan and Hon [5]. The Tanh method was successfully applied to nonlinear evolution equations [6, 7], and so on. The extended Tanh method was presented in [8] to solve breaking solitary equation. Wazwaz summarized the main steps introduced for using this method as follows [9]. We consider first a general form of nonlinear partial differential equation involving the two variables In this paper we only discuss the traveling wave solutions. (1) To find the traveling wave solution of (2.1), make the transformation where are constants to be determined later. From this reason, we use the following changes: and so on for the other derivates. Using (2.3) changes the NLPDE (2.1) to an ODE £¿ (2) If all terms of the resulting ODE contain
%U http://www.hindawi.com/journals/mpe/2010/935873/