Tempered stable Lévy motion and transient super-diffusion

✍ Scribed by Boris Baeumer; Mark M. Meerschaert

• Book ID: 104006959
• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 900 KB
• Volume: 233
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2009.10.027

✦ Synopsis

The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of powerlaw 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.