Abstract

The theory of anomalous diffusion is well-established and leads to the fractional PDEs for number densities. Despite the progress in understanding the anomalous transport most work has been concentrated on the passive density of the particles, and comparatively little is known about the interaction of anomalous transport with chemical reactions.

This work is intended to address this issue by utilising the random walk techniques. The main aim is to incorporate the nonlinear reaction terms into non-Markovian Master equations for a continuous time random walk (CTRW). We derive nonlinear evolution 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 of KPP -type. We find an explicit expression for the speed of a propagating front in the case of subdiffusive transport.