The (G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics

Abstract

I the present paper, we construct the traveling wave solutions involving parameters of the combined Korteweg-de Vries–modified Korteweg-de Vries equation, the reaction-diffusion equation, the compound KdV–Burgers equation, and the generalized shallow water wave equation by using a new approach, namely, the (G′/G)-expansion method, where G=G(ξ) satisfies a second order linear ordinary differential equation. When the parameters take special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions.

The Backlund Transformation of the Generalized Riccati Equation and Its Applications to the Nonlinear KPP Equation

“…Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equations. In recent decades, many effective methods have been established to obtain exact solutions of nonlinear PDEs, such as the inverse scattering transform [1], the Hirota method [2], the truncated Painlevé expansion method [3], the Bäcklund transform method [1,4,5], the exp-function method [6][7][8], the simplest equation method [9,10], the Weierstrass elliptic function method [11], the Jacobi elliptic function method [12][13][14], the tanh-function method [15,16], the ( / G) G′ expansion method [17][18][19][20][21][22], the modified simple equation method [23][24][25][26], the Kudryashov method [27][28][29], the multiple exp-function algorithm method [30,31], the transformed rational function method [32], the Frobenius decomposition technique [33], the local fractional variation iteration method [34], the local fractional series expansion The objective of this article is to use the Bäcklund transformation of the generalized Riccati equation to construct new exact traveling wave solutions of the following nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation [22,26,46]. [22] have discussed Equation (1.1) using the ( / G) G′ -expansion method and found its ex...…”

Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation

“…When a NLEE is analysed, one of the most important question is the construction of the exact solutions for equation [1]. In the open literature, quite a few methods for obtaining explicit travelling and solitary wave solutions to NLEEs have been suggested such as the inverse scattering method [2], the bilinear transformation method [3], the tanh-sech method [4,5], the extended tanh method [6,7], the sine-cosine method [8][9][10], the homogeneous balance method [11,12], the pseudo spectral method [13], the G ′ /G -expansion method [14][15][16], exp-function method [17], variational iteration method [18], homotopy perturbation method [19], the Jacobi elliptic function method [20], Lie group analysis method [21] and so on.…”

Travelling wave solutions of the generalized Zakharov-Kuznetsov equation via the extended generalized Riccati equation mapping method

“…This method obtains travelling wave solution of NLPDE by using the second order differential equation given by as an auxiliary equation. Zayed and Khaled, [17] applied the expansion method to construct travelling wave solution of the combined KdV-mKdV equation, the reaction-diffusion equation, the KdV-Burgers equation and the generalized shallow water wave equation. Abdollahzadeh et al, [18] employed the expansion method for the exact travelling wave solution of the Benjamin-Bona-Mahony-Burgers equation.…”