Non-Fickian delay reaction-diffusion equations: theoretical and numerical study

Abstract

The Fisher’s equation is established combining the Fick’s law for the flux and the mass conservation law. Assuming that the reaction term depends on the solution at some past time, a delay parameter is introduced and the delay Fisher’s equation is obtained. Modifying the Fick’s law for the flux considering a temporal memory term, integro-differential equations of Volterra type were introduced in the literature. In these paper we study reaction-diffusion equations obtained combining the two modifications: a temporal memory term in the flux and a delay in the reaction term. The delay integro-differential equations, also known as delay Volterra integrodifferential equations, are studied in the theoretical view point: stability estimates are established. Numerical methods which mimic the theoretical models are studied. Numerical experiments illustrating the established results are also included.