Tempered stable Lévy motion and transient super-diffusion

• 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
• DOI: 10.1016/j.cam.2009.10.027

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.