Computational Fractional Dynamical Systems

22Tanh Method

22.1 Introduction

Nonlinear phenomena have an essential role in various scientific fields. Exact solutions of these nonlinear phenomena are obtained using several approaches, including the tanh method, the inverse scattering method, Hirota's bilinear methodology, and the truncated Painleve expansion. Among them, the tanh method (Malfliet 1992; Malfliet 1996a; Malfliet 1996b; Wazwaz 2004; Wazwaz 2005; Wazwaz 2008) is a robust method for finding the exact traveling wave solutions. Huibin and Kelin (1990) proposed a power series in tanh as a solution and directly substituted this expansion into a higher order KdV equation. The coefficients of the power series were obtained from the resulting algebraic equations. The tanh method is one of the most straightforward technique for getting exact solutions to nonlinear diffusion equations (Khater et al. 2002). Various forms of the tanh method have been developed, and then a power series in tanh was utilized as an ansatz to get analytical solutions of certain nonlinear evolution equations (Malfliet 1996a).

In order to reduce the complexity of the tanh method, Malfliet (1992) modified the tanh approach by introducing tanh as a new variable. After that, a straightforward analysis was performed to ensure that the approach might be applicable to a wide range of equations (Khater et al. 2002). Later, in (Malfliet 1992; Malfliet 1996a; Malfliet 1996b), this approach was improved by adding the boundary conditions into the series ...