Tempered stable Lévy motion and transient super-diffusion

The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of power-law jumps. The tempered stable Lévy motion uses exponential tempering to cool these jumps. A tempered fractional diffusion...

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Bibliographic Details
Published in:Journal of computational and applied mathematics Vol. 233; no. 10; pp. 2438 - 2448
Main Authors: Baeumer, Boris, Meerschaert, Mark M.
Format: Journal Article
Language:English
Published: Kidlington Elsevier B.V 15.03.2010
Elsevier
Subjects:
Exact sciences and technology
Fractional derivatives
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Particle tracking
Power law
Real functions
Sciences and techniques of general use
Truncated power law
Fractional derivatives
Particle tracking
Power law
Truncated power law
Numerical analysis
Applied mathematics
Crank Nicolson method
Diffusion equation
Finite difference method
ISSN:0377-0427, 1879-1778
Online Access:Get full text
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Summary:The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of power-law jumps. The tempered stable Lévy 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 Lévy stables is presented to facilitate particle tracking codes.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2009.10.027