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Numerical solution of the space fractional Fokker-Planck equation. (English) Zbl 1036.82019

Summary: 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 \(\alpha\)-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 \(\alpha\) 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 \(\alpha\)-stable densities. We will show that the numerical method of SFFPE is able to more accurately model these heavy-tailed motions.

MSC:

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
26A33 Fractional derivatives and integrals
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