Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation

Abstract

The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order in (0,1). In this work, an explicit finite-difference scheme for TFDE is presented. Discrete models of a non-Markovian random walk are generated for simulating random processes whose spatial probability density evolves in time according to this fractional diffusion equation. We derive the scaling restriction of the stability and convergence of the discrete non-Markovian random walk approximation for TFDE in a bounded domain. Finally, some numerical examples are presented to show the application of the present technique.