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.

Using delay differential equations to induce alternans in a model of cardiac electrophysiology

“…Delays arise naturally in many settings in which sensing or reacting to changes in the system does not occur instantaneously. Some studies (Branco et al, 2010;Burić and Todorović, 2003;Campbell, 2007) have considered the effects of delays in reaction-diffusion systems, including models based on the FitzHugh-Nagumo equations.…”

Stability of Exact and Discrete Energy for Non-Fickian Reaction-Diffusion Equations with a Variable Delay

“…This pathologic behavior is not presented in the physical phenomena but introduced by some mathematical models (cf. [13]). In order to avoid the unphysical behavior, certain memory effects are taken into account in the mathematical models.…”

“…In order to avoid the unphysical behavior, certain memory effects are taken into account in the mathematical models. In [13][14][15], the time memory flux is proposed by assuming that a flux observed at some time should have something to do with the gradient of the solution at some past time; that is, ( , + ) = − 2 ( , ) .…”

“…In particular, some continuous and discrete energy estimates, which are determined by the 2 -norm of the solution and the estimates for the 2 -norm of the past in time of its gradient, were derived by Branco et al in [13]. However, the derived energy estimates rely on time.…”

Analytical solution for coupled non-Fickian diffusion-thermoelasticity and thermoelastic wave propagation analysis

“…In non-Fickian theory of diffusion, it is assumed the molar concentration is propagated with finite speed. There are some published works in which the non-Fickian diffusion problems were studied using numerical methods [7][8][9]. Kuang [3,10] proposed an inertial entropy concept and an inertial concentration or chemical potential concept.…”