New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp‐Function Method

Abstract

We construct new analytical solutions of the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation by the Exp-function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp-function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher-dimensional nonlinear partial differential equation.

A Novel (G<i>'</i>/G)-Expansion Method and its Application to the Space-Time Fractional Symmetric Regularized Long Wave (SRLW) Equation

“…With the development of symbolic computation software, likeMaple, many numerical and analytical methods to search for exact solutions of NLEEs have attracted more attention. As a result, the researchers developed and established many methods, for example, Cole-Hopf transformation [8], Tanh-function method [9][10][11][12][13], Inverse scattering transform method [14], Hirota method [15], Backlund transform method [16], Variational iteration method [17,18], Exp-function method [19][20][21][22][23], Extended tanh-method [24][25][26], Homogeneous balance method [27,28] and F-expansion method [29,30] are used for searching the exact solutions. Lately, Wang et al [31] introduced a direct and concise method, called (G'/G)-expansion method and demonstrated that it is a powerful method for seeking analytic solutions of NLEEs.…”

Application of an Enhanced (G&rsquo;/G)-Expansion Method to Find Exact Solutions of Nonlinear PDEs in Particle Physics

“…Many powerful method have been obtainable for instance the exp(-Φ(ξ))-expansion method (Khan et al, 2013a;Islam et al, 2014); the jacobi elliptic function method (Ali, 2011); the homogeneous balance method (Wang, 1995;Zayed et al, 2004); the modified simple equation method (Jawad et al, 2010;Khan and Akbar, 2013b;Zayed and Ibrahim, 2012;Akter and Akbar, 2015); the (G′/G)-expansion method (Wang et al, 2008;Zayed, 2010;Akbar et al, 2012b;Zayed and Gepreel, 2009;Akbar and Ali, 2011;Shehata, 2010;Akbar et al, 2012a;Mirzazadeh et al, 2014;; the improve (G′/G)-expansion method (Zhang et al, 2010); the extended(G′/G)-expansion method (Roshid et al, 2014a;; the generalized (G′/G)-expansion method 2014c); the novel (G′/G)-expansion method (Hafez et al, 2014); the homotopy perturbation method (Mohyud-Din et al, 2011a;2011b;2011c); the variational method (He, 1997;Abbasbandy, 2007;Arife and Yildirim, 2011;Abdou and Soliman, 2005); the exp-function method (Akbar and Ali, 2012;He and Wu, 2006;Naher et al, 2012); the truncated painleve expansion method (Weiss et al, 1983); the asymptotic method (He, 2008); the Hirota's bilinear transformation method (Hirota, 1973;Hirota and Satsuma, 1981); the tanhfunction method (Abdou, 2007;…”

Some new solutions of the (1+1)-dimensional PDE via the improved (G′/G)-expansion method

“…In recent years, several methods have been established for investigating nonlinear PDEs to obtain exact solutions. For example, the Hirota's bilinear transformation method [1], the B cklund transformation method [2], the Jacobi elliptic function expansion method [3], the generalized Riccati equation method [4,5], the homogeneous balance method [6], the F-expansion method [7], the Exp-function method [8][9][10][11].…”