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

Abstract

“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.