The auxiliary equation approach for solving reaction-diffusion equations

Abstract

The paper concerns the auxiliary equation method and proposes an approach for finding the most general nonlinear term that can generalize a nonlinear differential equation, so that it keeps solutions expressed in terms of the same auxiliary equation. More precisely we will consider the second order reaction-diffusion equations and we will find the most general nolinear term of this type of equations, for which the solutions can be expressed in terms of the Riccati equation. The procedure is exemplified on Fitzhugh-Nagumo, Dodd-Bullough-Mikhailov, and Klein-Gordon models, seen as reaction-diffusion equations.