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Publication Detail
Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation
  • Publication Type:
    Journal article
  • Publication Sub Type:
    Journal Article
  • Authors:
    Fulger D, Scalas E, Germano G
  • Publication date:
    25/02/2008
  • Journal:
    Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
  • Volume:
    77
  • Issue:
    2
  • Status:
    Published
  • Print ISSN:
    1539-3755
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. © 2008 The American Physical Society.
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