Numerical solution of the space fractional Fokker-Planck equation

Liu, Fawang, Anh, Vo V., & Turner, Ian W. (2004) Numerical solution of the space fractional Fokker-Planck equation. Journal of Computational and Applied Mathematics, 166(1), pp. 209-219.

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Abstract

The traditional second-order Fokker-Planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of Brownian motion. Densities of α-stable type have been used to describe the probability distribution of these motions. The resulting governing equation of these motions is similar to the traditional Fokker-Planck equation except that the order α of the highest derivative is fractional.In this paper, a space fractional Fokker-Planck equation (SFFPE) with instantaneous source is considered. A numerical scheme for solving SFFPE is presented. Using the Riemann-Liouville and Grünwald-Letnikov definitions of fractional derivatives, the SFFPE is transformed into a system of ordinary differential equations (ODE). Then the ODE system is solved by a method of lines. Numerical results for SFFPE with a constant diffusion coefficient are evaluated for comparison with the known analytical solution. The numerical approximation of SFFPE with a time-dependent diffusion coefficient is also used to simulate Lévy motion with α-stable densities. We will show that the numerical method of SFFPE is able to more accurately model these heavy-tailed motions.

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312 citations in Scopus
283 citations in Web of Science®
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ID Code: 10113
Item Type: Journal Article
Refereed: Yes
Additional Information: For more information, please refer to the journal’s website (see hypertext link) or contact the author.
Keywords: Fractional derivative, Fokker–Planck equation, α, stable densities, Lévy motion, Heavy, tailed motions
DOI: 10.1016/j.cam.2003.09.028
ISSN: 0377-0427
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Copyright Owner: Copyright 2004 Elsevier
Deposited On: 12 Oct 2007 00:00
Last Modified: 01 Mar 2012 09:27

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