Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts

Abstract

The main aim of the paper is to incorporate the non-linear kinetic term into non-Markovian transport equations described by a continuous time random walk (CTRW) with non-exponential waiting time distributions. We consider three different CTRW models with reactions. We derive new non-linear Master equations for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions. We apply these equations to the problem of front propagation in the reaction-transport systems with Kolmogorov-Petrovskii-Piskunov (KPP) kinetics and anomalous diffusion. We have found an explicit expression for the speed of a propagating front in the case of subdiffusive transport.

Fronts in anomalous diffusion–reaction systems

“…Nonlinear generalizations of equation (3.8) have been suggested by Froemberg et al [60] and Fedotov [61]. In the case of nonlinear dependence of reaction rates on reactant densities, it is typically impossible to write a closed system of equations for concentrations u i (x, t).…”

“…Surprisingly, solutions with a constant velocity have been found also in models without 'rejuvenation' [29,60,61]. The minimum front velocity can be either finite [61] (for reaction A + B → B + 2A) or zero [60] (for reaction A + B → 2A).…”

“…The crucial point is that A-type particles propagating into the region occupied by B-particles are always 'young' at the leading edge of the front, while B-type particles age continuously. In the model of Fedotov [61], the produced particles inherit the age of A-particles, and their mobility remains high. In the model of Froemberg et al [60], the produced particles inherit the age of B-particles, and their mobility decreases with time.…”

“…The minimum front velocity can be either finite [61] (for reaction A + B → B + 2A) or zero [60] (for reaction A + B → 2A). A subtle difference between the abovementioned models has been considered by Shkilev [63].…”

Mathematical Modelling of Subdiffusion-reaction Systems

“…Here the Laplace transform of the kernel,K(s), is proportional to sw(s)/[1 −w(s)], wherew(s) is the Laplace transform of the waiting time pdf. A number of nonlinear subdiffusion-reaction models have been derived by Fedotov [47]. The nonlinearity of the reaction term in the form R(u) = r(u)u can be taken into account by the following modification of equation (3.3):…”