The tanh function method for solving some important non-linear partial differential equations

Method for Studying the Multisoliton Solutions of the Korteweg-de Vries Type Equations

“…It is well known the KdV equations describe the unidirectional propagation of shallow water waves and a number of generalizations; for example, + ( + ) + = 0 (is given by [2]), The KdV equations extended to several physical problems such as long internal waves in a density stratified ocean and acoustic waves on a crystal lattice. In recent years many efficient methods of finding the traveling wave solutions were developed such as Infinite Series method [8], Backlund transformation method [9], Darboux transformation [10], tanh method [11,12], extended tanh function method [4], modified and extended tanh function method [13], the generalized hyperbolic function [14], the variable separation method, first integral method, and exp-function method [15]. A number of papers were devoted to the problems of the asymptotic solutions of the Korteweg-de Vries equation [16].…”

The Tanh Methods for the Hirota Equations

“…The availability of symbolic computations such as Mathematica program has popularized direct seeking for exact solutions of nonlinear equations. Therefore, exact solution methods of nonlinear evolution equations have become more and more important resulting in methods like the tanh method [1][2][3], extended tanh function method [4], the modified extended tanh function method [5], the generalized hyperbolic function [6].Most of exact solutions have been obtained by these methods, including the solitary wave solutions, shock wave solutions, periodic wave solutions, and the like.…”

The Study of Heat Transfer Phenomena Using PM for Approximate Solution with Dirichlet and Mixed Boundary Conditions

“…The assumption that the nonlinear part is small compared to the linear is considered as a disadvantage of the method. There are other modern alternatives to find approximate solutions to the differential equations that describe some nonlinear problems such as those based on: variational approaches [5][6][7]29], tanh method [8], exp-function [9, 10], Adomian's decomposition method [11][12][13][14][15][16], parameter expansion [17], homotopy perturbation method [3,4,16,[18][19][20][21][22][23][24][25][26][27][28][29][31][32][33][34][35][36]38,42], homotopy analysis method [30], Homotopy asymptotic method [1], and perturbation method [39,41] among many others. Also, a few exact solutions to nonlinear differential equations have been reported occasionally [40].…”