Exact Travelling Wave Solutions for Generalized Forms of the Nonlinear Heat Conduction Equation

“The most incomprehensible thing about the world is that it is at all comprehensible” (Albert Einstein), but the question is how do we fully understand incomprehensible things? Nonlinear science provides some clues in this regard (He, 2009). The world around us is inherently nonlinear. For instance, nonlinear evolution equations (NLEEs) are widely used as models to describe complex physical phenomena in various fields of sciences, especially in fluid mechanics, solid-state physics, plasma physics, plasma waves, and biology. One of the basic physical problems for these models is to obtain their travelling wave solutions. In particular, various methods have been utilized to explore different kinds of solutions of physical models described by nonlinear partial differential equations (PDEs). For instance, in the numerical methods, stability and convergence should be considered, so as to avoid divergent or inappropriate results. However, in recent years, a variety of effective analytical and semi-analytical methods have been developed to be used for solving nonlinear PDEs, such as the variational iteration method (VIM) (He, 1998; He et al., 2010), the homotopy perturbation method (HPM) (He, 2000, 2006), the homotopy analysis method (HAM) (Abbasbandy, 2010), the tanh-method (Fan, 2002; Wazwaz, 2005, 2006), the sine-cosine method (Wazwaz, 2004), and others. Likewise, He and Wu (2006) proposed a straightforward and concise method called the Exp-function method to obtain the exact solutions of NLEEs. The method, with the aid of Maple or Matlab, has been successfully applied to many kinds of NLEE (He & Zhang, 2008; Kabir & Khajeh, 2009; Borhanifar & Kabir, 2009, 2010; Borhanifar et al., 2009; Kabir et al., 2011). Lately, the (G′/G)expansion method, first introduced by Wang et al. (2008), has become widely used to search for various exact solutions of NLEEs (Bekir & Cevikel, 2009; Zhang et al., 2009; Zedan, 2010; Kabir et al., 2011). The results reveal that the two recent methods are powerful techniques for solving nonlinear partial differential equations (NPDEs) in terms of accuracy and efficiency. This is important, since systems of NPDEs have many applications in engineering.

The generalized forms of the nonlinear heat conduction equation can be given as ()  The heat equation is an important partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time. The heat equation is a consequence of Fourier's law of cooling. In this chapter, we consider the heat equation with a nonlinear power-law source term. The equations (1.1) and (1.2) describe one-dimensional and two-dimensional unsteady thermal processes in quiescent media or solids with the nonlinear temperature dependence of heat conductivity. In the above equations, u= u(x,y,t) is temperature as a function of space and time; t u is the rate of change of temperature at a point over time; xx u and yy u are the second spatial derivatives (thermal conductions) of temperature in the x and y directions, respectively; also x u and y u are the temperature gradient. Many authors have studied some types of solutions of these equations. Wazwaz (2005) used the tanh-method to find solitary solutions of these equations and a standard form of the nonlinear heat conduction equation (when 3 n  in Eq. (1.1)). Also, Fan (2002) applied the solutions of Riccati equation in the tanh-method to obtain the travelling wave solution when 2 n  in Eq. (1.1). More recently, Kabir et al. (2009) implemented the Exp-function method to find exact solutions of Eq. (1.1), and obtained more general solutions in comparison with Wazwaz's results. Considering all the indispensably significant issues mentioned above, the objective of this paper is to investigate the travelling wave solutions of Eqs. (1.1) and (1.2) systematically, by applying the (G'/G)-expansion and the Exp-function methods. Some previously known solutions are recovered as well, and, simultaneously, some new ones are also proposed.

The (G'/G)-expansion method
Suppose that a nonlinear PDE, say in two independent variables x and t, is given by (, , , , , , )  (2.1) or in three independent variables x, y and t, is given by (, , , , , , , , , ) and so on. Here, the prime denotes the derivative with respective to  .
To determine u explicitly, we take the following four steps: Step 1. Determine the integer m by substituting Eq. (2.7) along with Eq. (2.8) into Eq. (2.5) or (2.6), and balancing the highest-order nonlinear term(s) and the highest-order partial derivative.
Step 2 Step 3. Solve the system of algebraic equations obtained in Step 2, for 0 ,, kc and i  , for 1, 2, ... , im  , by use of Maple.
Step 4. Use the results obtained in the above steps to derive a series of fundamental

The Exp-function method
According to the classic Exp-function method, it is assumed that the solution of ODEs (2.5) or (2.6) can be written as where ,, f gpand q are positive integers which are unknown, to be further determined, and n a and m b are unknown constants.

Application of the (G'/G)-expansion method
To get a closed-form analytic solution, we use the transformation (Kabir & Khajeh, 2009;Wazwaz, 2005)  (1 ) According to Step 1 where 0  and 1  , are constants which are unknown, to be determined later.
where  and  are arbitrary constants.
By using Eq. (3.8), expression (3.7) can be written as Substituting the general solution of (2.9) into Eq. (3.9), we get the generalized travelling wave solution as follows: inserting Eq. (3.10) into Eq. (3.3), it yields the following exact solution of Eq. (1.1): Similarly, if we set 2 0 C  and 1 0 C  in the general solution (3.28), we arrive at the same solutions (3.18) and (3.19), respectively.

Application of the Exp-function method
By the same manipulation as illustrated above, we have the following sets of solutions: Case 1.

www.intechopen.com
We note that if we set 00 ab  in Eq. (3.48), we can recover the solution (3.58).
and by inserting Eq. (3.60) into (3.3), we get the generalized solitary wave solution of (1.1) as

Application of the (G'/G)-expansion method
By the same manipulation as illustrated in Section 3.1, we obtain the following sets of solutions. (4.5) in which 1 C and 2 C are arbitrary parameters that can be determined by the related initial and boundary conditions. Now, to obtain some special cases of the above general solution, we set 2 0 C  ; then (4.5) leads to   We note that, if we set 2 0 C  and 1 0 C  in the general solution (4.13), we can recover the solutions (4.6) and (4.7), respectively.
Case B-2. In particular, if we take 2 0 C  and 1 0 C  in the general solution (4.15), we arrive at the same solutions (4.10) and (4.11), respectively.

Application of the Exp-function method
By the same manipulation as illustrated in Section 3.2, we obtain the following sets of solutions. a  is an arbitrary parameter which can be determined by the initial and boundary conditions.