Exact solutions of Fisher and Burgers equations with finite transport memory

Abstract

The Fisher and Burgers equations with finite memory transport, describing reaction–diffusion and convection–diffusion processes, respectively have recently attracted a lot of attention in the context of chemical kinetics, mathematical biology and turbulence. We show here that they admit exact solutions. While the speed of the travelling wavefront is dependent on the relaxation time in the Fisher equation, memory effects significantly smoothen out the shock wave nature of the Burgers solution, without any influence on the corresponding wave speed. We numerically analyse the ansatz for the exact solution and show that for the reaction–diffusion system the strength of the reaction term must be moderate enough not to exceed a critical limit to allow a travelling wave solution to exist for appreciable finite memory effect.