Linearizing transformations to a generalized reaction–diffusion system

Abstract

Linearizing nonlinear differential equations and finding exact solutions has been an interesting subject in the theory of differential equations. For many nonlinear differential equations, so far no general treatment on linearizing transformations and obtaining exact solutions has been formulated. In this article we use the generalized linearizing transformation to derive the first integral of a generalized reaction–diffusion equation with higher order nonlinear rate of growth, which enables us to linearize the associated second-order nonlinear ordinary differential equation to a linear case. Under certain parametric conditions we can also find travelling wave solutions explicitly for the generalized reaction–diffusion equation through using the derived first integral.

Keywords:

• reaction–diffusion system
• linearizing transformation
• first integral
• travelling wave
• Fisher equation
• general solution

AMS Subject Classifications:

• 35K57
• 34A05

Acknowledgements

The main content has been present at the NCTS 2009 Interdisciplinary Conference on Applied Analysis and Mathematics, 11–13 May 2009, National Tsing-Hua University, Hsinchu, Taiwan. The work is supported by UTPA Faculty Research Grant 119100. This work is also partially supported by TUTE Research Grant 2006518.