Symbolic Computations and Exact and Explicit Solutions of Some Nonlinear Evolution Equations in Mathematical Physics

Abstract

With the aid of symbolic computation system Mathematica, several explicit solutions for Fisher's equation and CKdV equation are constructed by utilizing an auxiliary equation method, the so called G /G-expansion method, where the new and more general forms of solutions are also constructed. When the parameters are taken as special values, the previously known solutions are recovered.

Variational approach, soliton solutions and singular solitons for new coupled ZK system

“…So we should search for a mathematical algorithm to discover the exact solutions of nonlinear partial differential equations. In recent years, powerful and efficient methods explored to find analytic solutions of nonlinear equations have drawn a lot of interest by a variety of scientists, such as Adomian decomposition method [2], the homotopy perturbation method [3,4], some new asymptotic methods searching for solitary solutions of nonlinear differential equations, nonlinear differential-difference equations and nonlinear fractional differential equations using the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform and ancient Chinese mathematics [4], the variational iteration method [5,6] which is used to introduce the definition of fractional derivatives [7,4], the He's variational approach [8], the extended homoclinic test approach [9,10], homogeneous balance method [11][12][13][14], Jacobi elliptic function method [15][16][17][18], Băclund transformation [19,20], G ′ /G expansion method for nonlinear partial differential equation [21,22], and fractional differential-difference equations of rational type [23][24][25] It is important to point out that a new constrained variational principle for heat conduction is obtained recently by the semi-inverse method combined with separation of variables [26], which is exactly the same with He-Lee's variational principle [27]. A short remark on the history of the semi-inverse method for establishment of a generalized variational principle is given in [28].…”

The discrete (G′/G)‐expansion method applied to the differential‐difference Burgers equation and the relativistic Toda lattice system

“…Right after their pioneer work, the (G /G)-expansion method became popular in the research community, and a number of studies refining the initial idea have been published [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. The value of the (G /G)-expansion method is that one treats nonlinear problems by essentially linear methods.…”

Application of the Exp-function method to the (2+1)-dimensional Boiti–Leon–Pempinelli equation using symbolic computation

“…Thus, in the last four decades or so, many different methods have been presented in the open literature to look for exact solutions of these equations. To mention some, Darboux transformation [33], Hirota bilinear method [27], inverse scattering transformation [2], symmetry method [9], Weierstrass function method [43], Jacobi elliptic function method [29], Sine-Cosine function [40], Tanh-Coth function [32], F-expansion method [1], homotopy perturbation method [15,16,23], variational iteration method [14,22,38], Adomian decomposition method [3,17,18], (G /G)-expansion method [4,6,7,35,36,39], and so forth. However, most of the methods have some restrictions for application purposes.…”