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Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations

Yang, Qianqian, Liu, Fawang, & Turner, Ian (2009) Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations. Physica Scripta, T136, pp. 1-7.

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Abstract

Fractional Fokker–Planck equations have been used to model several physical situations that present anomalous diffusion. In this paper, a class of time- and space-fractional Fokker–Planck equations (TSFFPE), which involve the Riemann–Liouville time-fractional derivative of order 1-α (α(0, 1)) and the Riesz space-fractional derivative (RSFD) of order μ(1, 2), are considered. The solution of TSFFPE is important for describing the competition between subdiffusion and Lévy flights. However, effective numerical methods for solving TSFFPE are still in their infancy.

We present three computationally efficient numerical methods to deal with the RSFD, and approximate the Riemann–Liouville time-fractional derivative using the Grünwald method. The TSFFPE is then transformed into a system of ordinary differential equations (ODE), which is solved by the fractional implicit trapezoidal method (FITM). Finally, numerical results are given to demonstrate the effectiveness of these methods. These techniques can also be applied to solve other types of fractional partial differential equations.

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ID Code: 37945
Item Type: Journal Article
Keywords: Computational physics, Statistical physics and nonlinear systems
DOI: 10.1088/0031-8949/2009/T136/014026
ISSN: 0031-8949
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) > Numerical Solution of Differential and Integral Equations (010302)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Past > Schools > Mathematical Sciences
Copyright Owner: Copyright 2009 The Royal Swedish Academy of Sciences
Deposited On: 14 Oct 2010 12:07
Last Modified: 01 Mar 2012 00:07

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