The Functional Expansion Approach for Solving NPDEs as a Generalization of the Kudryashov and G′/G Methods

Abstract

This paper presents the functional expansion approach as a generalized method for finding traveling wave solutions of various nonlinear partial differential equations. The approach can be seen as a combination of the Kudryashov and G′/G solving methods. It allowed the extension of the first method to the use of second order auxiliary equations, and, at the same time, it allowed non-standard G′/G-solutions to be generated. The functional expansion is illustrated here on the Dodd–Bullough–Mikhailov model, using a linear second order ordinary differential equation as an auxiliary equation.

Painlevé Test, Phase Plane Analysis and Analytical Solutions of the Chavy–Waddy–Kolokolnikov Model for the Description of Bacterial Colonies

“…The pole value of the general solution is p = 4. Taking into account this value, we look for the solution of Equation ( 16) of the form (see [40][41][42][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68])…”

The auxiliary equation approach for solving reaction-diffusion equations

“…For example, following the choice of the auxiliary equation, the most used equations chosen to fulfill this role are the Riccati equation and the elliptic Jacobi equation [6], but many other equations have been also used. Following the form in which the solutions are sought, different approaches were proposed and intensively used in literature: sine-cosine method [7], tanh-method [8], Kudryashov method [9,10], the functional expansion method [11], exp-function method [12], (G ′ /G)-method [13], the attached flow approach [14], etc. In the following, a possible approach will be presented to establish the most general nonlinear equation compatible with a given auxiliary equation.…”

The degeneration of solitary wave solutions and interaction between lump and soliton solutions for the (2+1)-dimensional KdV equation

“…Through many years of research by many authors, some efficient methodologies are successfully advanced to solve exact solutions of NLPDEs. For instance, Hirota bilinear method [18], Darboux transformation [19][20][21], (G ′ /G)expansion method [22][23][24], three-wave approach [25][26][27], extended homoclinic test method [28][29][30], etc.…”