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

TL;DR

This paper introduces a Monte Carlo simulation method for uncoupled continuous-time random walks with Levy alpha-stable jumps and Mittag-Leffler waiting times, providing an efficient stochastic solution to space-time fractional diffusion equations.

Contribution

It develops a combined transformation method for Mittag-Leffler and Levy alpha-stable variables, enabling accurate and fast simulation of space-time fractional diffusion processes.

Findings

Efficient Monte Carlo method for fractional diffusion equations.

Accurate approximation of space- and time-fractional processes.

Comparable computational speed to standard diffusion simulations.

Abstract

We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Levy alpha-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 Levy alpha-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.