Tempered stable Lévy motion and transient super-diffusion

Authors:
Boris Baeumer;Mark M. Meerschaert
Affiliations:
Department of Mathematics & Statistics, University of Otago, Dunedin, New Zealand;Department of Statistics & Probability, Michigan State University, Wells Hall, E. Lansing, MI 48824, United States
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

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Abstract

The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Levy motion, representing the accumulation of power-law jumps. The tempered stable Levy motion uses exponential tempering to cool these jumps. A tempered fractional diffusion equation governs the transition densities, which progress from super-diffusive early-time to diffusive late-time behavior. This article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift. A temporal and spatial second-order Crank-Nicolson method is developed, based on a finite difference formula for tempered fractional derivatives. A new exponential rejection method for simulating tempered Levy stables is presented to facilitate particle tracking codes.