Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation
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.
- Publication:
-
Physical Review E
- Pub Date:
- February 2008
- DOI:
- 10.1103/PhysRevE.77.021122
- arXiv:
- arXiv:0707.2582
- Bibcode:
- 2008PhRvE..77b1122F
- Keywords:
-
- 02.50.Ng;
- 05.70.Ln;
- 02.70.Tt;
- 02.70.Uu;
- Distribution theory and Monte Carlo studies;
- Nonequilibrium and irreversible thermodynamics;
- Justifications or modifications of Monte Carlo methods;
- Applications of Monte Carlo methods;
- Condensed Matter - Statistical Mechanics
- E-Print:
- 7 pages, 5 figures, 1 table. Presented at the Conference on Computing in Economics and Finance in Montreal, 14-16 June 2007