A time-fractional generalised advection equation from a stochastic process

Abstract

A generalised advection equation with a time fractional derivative is derived from a continuous time random walk on a one-dimensional lattice, with power law distributed waiting times. We consider walks governed by a two-sided jump density and walks governed by a one-sided jump density. With the two-sided density, the particle can jump in both directions on the lattice, whereas with the one-sided density the particle cannot jump in one of these directions. The master equations describing the evolution of the probability density for the position of the particle are different for each of the jump densities. However in an advective limit both master equations limit to a common generalized advection equation with time fractional derivatives.We have also considered the stochastic processes in a discrete time setting, again arriving at different discrete time master equations for each of the jump densities. The discrete time master equations can be used to provide different numerical approximations to the solutions of the fractional generalized advection equation. The approximations allow us to compare the efficacy of the one-sided and two-sided densities.