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Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution... We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Lévy α -stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the stochastic solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter Mittag-Leffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag-Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for Lévy α -stable random variables and so far have not received as much attention; nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the Mittag-Leffler distribution. Combining the two methods, we obtain an accurate approximation of space- and time-fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review E American Physical Society (APS)

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

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References (86)

Publisher
American Physical Society (APS)
Copyright
Copyright © 2008 The American Physical Society
ISSN
1550-2376
DOI
10.1103/PhysRevE.77.021122
pmid
18352002
Publisher site
See Article on Publisher Site

Abstract

We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Lévy α -stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the stochastic solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter Mittag-Leffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag-Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for Lévy α -stable random variables and so far have not received as much attention; nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the Mittag-Leffler distribution. Combining the two methods, we obtain an accurate approximation of space- and time-fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes.

Journal

Physical Review EAmerican Physical Society (APS)

Published: Feb 1, 2008

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