TY  - JOUR
AB  - 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.
VL  - 23
IS  - 1
SN  - 0961-5539
DO  - 10.1108/09615531311289088
UR  - https://doi.org/10.1108/09615531311289088
AU  - Zielinski D.P.
AU  - Voller V.R.
ED  - X. Li
ED  - N. Massarotti
ED  - P. Nithiarasu
PY  - 2013
Y1  - 2013/01/01
TI  - A random walk solution for fractional diffusion equations
T2  - International Journal of Numerical Methods for Heat & Fluid Flow
PB  - Emerald Group Publishing Limited
SP  - 7
EP  - 22
Y2  - 2024/05/11
ER  -