A hybrid numerical treatment of nonlinear reaction-diffusion equations with memory: A prototypical Fisher-Kolmogorov-Petrovskii-Piskunov equation

Abstract

In reaction-diffusion systems with non-standard diffusion, the memory of the transport process causes a coupling of reaction and diffusion. A generalisation of the Fick's law has been suggested to account for this coupling. Furthermore, the resultant effects of the interplay of transport, memory and reaction lend themselves to some interesting physics which is still not well understood because the governing equation as well as the accompanying memory integral and nonlinear reaction terms are not always amenable to tidy analytic or numerical expressions. Hence, the derivation of a suitable governing integro-differential equation as well as the approximate solution demands a considerable level of attention'. The main focus of this work can be seen as a contribution towards this objective. In this report, we develop and apply a hybrid boundary integral-finite element - finite difference numerical procedure to investigate an integro-differential-Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) - type kinetics. We also focus on scalar evolution for cases where the reaction coefficient takes on relatively large values. Although we are still far from a rigorous mathematical analysis, it has been found that the numerical results obtained compared favourably with existing benchmark solutions. This not only validates the current numerical formulation but also justifies the physics of the resulting front propagation.