A random walk solution for fractional diffusion equations

Purpose - The purpose of this paper is to develop an alternative numerical approach for describing fractional diffusion in Cartesian and non-Cartesian domains using a Monte Carlo random walk scheme. The resulting domain shifting scheme provides a numerical solution for multi-dimensional steady state, source free diffusion problems with fluxes expressed in terms of Caputo fractional derivatives. This class of problems takes account of non-locality in transport, expressed through parameters representing both the extent and direction of the non-locality. Design/methodology/approach - The method described here follows a similar approach to random walk methods previously developed for normal (local) diffusion. The key differences from standard methods are: first, the random shifting of the domain about the point of interest with, second, shift steps selected from non-symmetric, power-law tailed, Lévy probability distribution functions. Findings - The domain shifting scheme is verified by comparing predictive solutions to known one-dimensional and two-dimensional analytical solutions for fractional diffusion problems. The scheme is also applied to a problem of fractional diffusion in a non-Cartesian annulus domain. In contrast to the axisymmetric, steady state solution for normal diffusion, a non-axisymmetric solution results. Originality/value - This is the first random walk scheme to utilize the concept of allowing the domain to undergo the random walk about a point of interest. Domain shifting scheme solutions of fractional diffusion in non-Cartesian domains provide an invaluable tool to direct the development of more sophisticated grid based finite element inspired fractional diffusion schemes.