Expanding the Tanh-Function Method for Solving Nonlinear Equations

Abstract

In this paper, using the tanh-function method, we introduce a new approach to solitary wave solutions for solving nonlinear PDEs. The proposed method is based on adding integration constants to the resulting nonlinear ODEs from the nonlinear PDEs using the wave transformation. Also, we use a transformation related to those integration constants. Some examples are considered to find their exact solutions such as KdV- Burgers class and Fisher, Boussinesq and Klein-Gordon equations. Moreover, we discuss the geometric interpretations of the resulting exact solutions

Optimized Direct Padé and HPM for Solving Equation of Oxygen Diffusion in a Spherical Cell

“…In order to reduce the amount of algebraic steps, it is proposed in this work, and for the first time, to expand the rational expression (21) and employ the Taylor series of order four:…”

“…There are several methods of solutions for solving nonlinear differential equations such as inverse scattering transformation [14,15], Darboux transformation [16,17], bilinear method [18,19], the tanh-function method [20][21][22], the variable separation approach [23][24][25], the symmetry method [26,27], sine-cosine method [28][29][30], Adomian decomposition method (ADM) [12,[31][32][33], and homotopy perturbation method (HPM) [34][35][36]. HPM is based on the use of power series of , which transforms a differential equation into a set of linear differential equations.…”

Solitary Wave Solutions of Some Coupled Nonlinear Evolution Equations

“…However, not all equations posed of these models are solvable. As a result, many new techniques have been successfully developed by diverse groups of mathematicians and physicists, such as the Hirota's bilinear transformation method [1,2], the tanh-function method [3,4], the extended tanh-method [5,6], the Exp-function method [7][8][9][10][11], the Adomian decomposition method [12], the F-expansion method [13], the auxiliary equation method [14], the Jacobi elliptic function method [15], the modified exp-function method [16], the (G'/G)-expansion method [17][18][19][20][21][22][23][24][25][26], the Weierstrass elliptic function method [27], the homotopy perturbation method [28][29][30], the homogeneous balance method [31,32], the modified simple equation method [33][34][35][36], the enhanced (G'/G)-expansion method [37], the exp(-Φ(ξ))-expansion method [38], the ansatz method [39], the functional variable method [40] and so on.…”

An efficient technique for higher order fractional differential equation

“…The investigation of travelling wave solutions (Shawagfeh 2002; Ray and Bera 2005; Yildirim et al 2011; Kilbas et al 2006; He and Li 2010; Momani and Al-Khaled 2005; Odibat and Momani 2007; Abdou 2007; Nassar et al 2011; Misirli and Gurefe 2011; Noor et al 2008; Ozis and Koroglu 2008; Wu and He 2007; Yusufoglu 2008; Zhang 2007; Zhu 2007; Wang et al 2008; Zayed et al 2004; Sirendaoreji 2004; Ali 2011; Liang et al 2011; He et al 2012; Jawad et al 2010; Zhou et al 2003; Yıldırım and Kocak 2009; Elbeleze et al 2013; Matinfar and Saeidy 2010; Ahmad 2014; Bongsoo 2009; Demiray and Pandir 2014, 2015; Lu 2012; Zayed and Amer 2014) of nonlinear evolution equations plays a significant role to look into the internal mechanism of nonlinear physical phenomena. Nonlinear fractional differential equations (FDEs) are a generalization of classical differential equations of integer order.…”

“…In recent years, considerable interest in fractional differential equations (He and Li 2010; Momani and Al-Khaled 2005; Odibat and Momani 2007) has been stimulated due to their numerous applications in different fields. However, many effective and powerful methods have been established and improved to study soliton solutions of nonlinear equations, such as extended tanh-function method (Abdou 2007), tanh-function method (Nassar et al 2011), Exp-function method (Misirli and Gurefe 2011; Noor et al 2008; Ozis and Koroglu 2008; Wu and He 2007; Yusufoglu 2008; Zhang 2007; Zhu 2007), ( G ’ / G )-expansion method (Wang et al 2008), homogeneous balance method (Zayed et al 2004), auxiliary equation method (Sirendaoreji 2004), Jacobi elliptic function method (Ali 2011), Weierstrass elliptic function method (Liang et al 2011), modified Exp-function method (He et al 2012), modified simple equation method (Jawad et al 2010), F-expansion method (Zhou et al 2003), homotopy perturbation method (Yıldırım and Kocak 2009), Fractional variational iteration method (Elbeleze et al 2013), homotopy analysis method (Matinfar and Saeidy 2010), Reduced differential transform method (Ahmad 2014), Generalized Kudryashov method for time-fractional differential equations (Demiray and Pandir 2014), The first integral method for some time fractional differential equations(Lu 2012; Zayed and Amer 2014), New solitary wave solutions of Maccari system (Demiray and Pandir 2015), and so on.…”