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Finite difference methods for two-dimensional fractional dispersion equation. (English) Zbl 1085.65080
Summary: 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. The authors 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 Grünwald 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.

65M06Finite difference methods (IVP of PDE)
26A33Fractional derivatives and integrals (real functions)
35G25Initial value problems for nonlinear higher-order PDE
65M12Stability and convergence of numerical methods (IVP of PDE)
35G30Boundary value problems for nonlinear higher-order PDE