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

Daniel Fulger1,*, Enrico Scalas2,†, and Guido Germano1,‡

  • 1Department of Chemistry and WZMW, Computer Simulation Group, Philipps-University Marburg, 35032 Marburg, Germany
  • 2Department of Advanced Sciences and Technology, Laboratory on Complex Systems, Amedeo Avogadro University of East Piedmont, Via Vincenzo Bellini 25 G, 15100 Alessandria, Italy

  • *daniel.fulger@staff.uni-marburg.de
  • †enrico.scalas@mfn.unipmn.it; URL: www.mfn.unipmn.it/∼scalas
  • ‡Corresponding author. guido.germano@staff.uni-marburg.de; URL: www.staff.uni-marburg.de/∼germano
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Phys. Rev. E 77, 021122 – Published 25 February, 2008

DOI: https://doi.org/10.1103/PhysRevE.77.021122

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.

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Phys. Rev. E 77, 021122– Published 25 February, 2008
  • Received 24 August 2007

© 2008 American Physical Society

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