Finite difference methods for two-dimensional fractional dispersion equation

Authors:
Mark M. Meerschaert;Hans-Peter Scheffler;Charles Tadjeran
Affiliations:
Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand;Department of Mathematics and Statistics, University of Nevada, Reno, USA;Department of Physics, University of Nevada, Reno, USA
Venue:
Journal of Computational Physics
Year:
2006

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Abstract

Fractional order partial differential equations, as generalizations of classical integer order partial differential equations, are increasingly used to model problems in fluid flow, finance and other areas of application. In this paper we discuss a practical alternating directions implicit method to solve a class of two-dimensional initial-boundary value fractional partial differential equations with variable coefficients on a finite domain. First-order consistency, unconditional stability, and (therefore) first-order convergence of the method are proven using a novel shifted version of the classical Grunwald finite difference approximation for the fractional derivatives. A numerical example with known exact solution is also presented, and the behavior of the error is examined to verify the order of convergence.