The Modified (G'/G)-Expansion Method for Nonlinear Evolution Equations

Abstract

A modified (G /G)-expansion method is proposed to construct exact solutions of nonlinear evolution equations. To illustrate the validity and advantages of the method, the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation is considered and more general travelling wave solutions are obtained. Some of the obtained solutions, namely hyperbolic function solutions, trigonometric function solutions, and rational solutions contain an explicit linear function of the variables in the considered equation. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

The (G′/G,1/G)-Expansion Method and Its Applications to Find the Exact Solutions of Nonlinear PDEs for Nanobio

“…We close this case with the remark that our results (27)- (29) and (31) are new and are not reported elsewhere.…”

“…In recent years, investigations of exact solutions to nonlinear partial differential equations (PDEs) play an important role in the study of nonlinear physical phenomena. Many powerful methods for finding these exact solutions have been presented, such as the inverse scattering method [1], the Hirota bilinear transform method [2], the truncated Painleve expansion method [3][4][5][6], the Backlund transform method [7,8], the exp-function method [9][10][11][12][13], the tanh-function method [14][15][16][17], the Jacobi elliptic function expansion method [18][19][20], the ( / )-expansion method [21][22][23][24][25][26][27][28][29][30], the modified ( / )-expansion method [31], the ( / , 1/ )-expansion method [32][33][34][35], the modified simple equation method [36], the multiple exp-function algorithm method [37], the transformed rational function method [38], the local fractional variation iteration method [39], and the local fractional series expansion method [40]. Further exact solutions to some reallife physical problems were already given in the recent articles [41]…”

The (G'/G)-Expansion Method for the Sine-Gordon Equation, Sinh-Gordon Equation and Liouville Equaiton

“…These equations have been investigated by different methods, and some exact solutions are derived [5][6][7][8]. However, there are still many other exact solutions to be found.…”

The(G'/G,1/G)-Expansion Method and Its Applications for Solving Two Higher Order Nonlinear Evolution Equations

“…In the recent years, investigations of exact solutions to nonlinear PDEs play an important role in the study of nonlinear physical phenomena. Many powerful methods have been presented, such as the inverse scattering method [1], the Hirota bilinear transform method [2], the truncated Painleve expansion method [3][4][5][6], the Backlund transform method [7,8], the exp-function method [9][10][11][12][13], the tanhfunction method [14][15][16][17], the Jacobi elliptic function expansion method [18][19][20], the ( / )-expansion method [21][22][23][24][25][26][27][28][29][30], the modified ( / )-expansion method [31], and the ( / , 1/ )-expansion method [32][33][34]. The key idea of the one variable ( / )-expansion method is that the exact solutions of nonlinear PDEs can be expressed by a polynomial in one variable ( / ) in which = ( ) satisfies the second order linear ODE ( ) + ( ) + ( ) = 0, where and are constants and = / .…”