A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms

Direct method for solving nonlinear strain wave equation in microstructure solids

“…The search for exact traveling wave solutions to nonlinear evaluation equations plays very important role in the study of these physical phenomena. In recent years, the exact solutions of nonlinear partial differential equation have been investigated by many authors (Ablowitz and Clarkson, 1991;Rogers and Shadwick, 1982;Matveev and Salle, 1991;Li and Chen, 2003;Conte and Musette, 1992;Ebaid and Aly, 2012;Gepreel, 2014;Cariello and Tabor, 1991;Fan, 2000;Fan, 2002;Wang and Li, 2005;Abdou, 2007;Wu and He, 2006;Wu and He, 2008;Li and Wang, 2007;Zheng, 2012;Triki and Wazwaz, 2014;Bibi and Mohyud-Din, 2014;Yu-Bin and Chao, 2009;Zayed and Gepreel, 2009;He, 2006;Gepreel, 2011;Adomian, 1988;Wazwaz, 2007;Liao, 2010;Gepreel and Mohamed, 2013;Wang et al, 2008;Yan, 2003a) (Ablowitz and Clarkson, 1991), the Backlund transform (Rogers and Shadwick, 1982), Darboux transform (Matveev and Salle, 1991), the generalized Riccati equation (Li and Chen, 2003;Conte and Musette, 1992), the Jacobi elliptic function expansion method (Ebaid and Aly, 2012;Gepreel, 2014), Painlev´e expansions method (Cariello and Tabor, 1991), the extended Tang-function method …”

The (F/G)-Expansion Method and Travelling Wave Solutions of Nonlinear Evolution Equations

“…During the past four decades or so, the many researchers are interested to find powerful and efficient methods for analytic solutions of nonlinear equations. Many powerful methods to obtain exact solutions of nonlinear evolution equations have been constricted and developed such as the inverse scattering transform in [1], the Backlund/Darboux transform in [2][3][4], the Hirota's bilinear operators in [5], the truncated Painleve expansion in [6], the tanh-function expansion and its various extension in [7][8][9], the Jacobi elliptic function expansion in [10,11], the F-expansion in [12][13][14][15], the sub-ODE method in [16][17][18][19], the homogeneous balance method in [20][21][22], the sine-cosine method in [23,24] the rank analysis method in [25], the ansatz method in [26][27][28], the expfunction expansion method in [29] and so on, but there is no unified method that can be used to deal with all types of nonlinear evolution equations. )evolution equation with variable coefficients.…”

Solitary wave solutions for a generalized KdV–mKdV equation with distributed delays

“…And many powerful methods to construct exact solutions of KdV have been established and developed. Among these methods we mainly cite, for example, the bifurcation method of dynamical systems [8], (G /G)-expansion method [9][10][11] and the sub-ODE method [12], Lie group theoretical methods [13] and so on. In this paper, we will use the geometrical singular perturbation theory and the linear chain trick to investigate solitary wave solutions of the generalized KdV-mKdV equation.…”