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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.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
35G25 Initial value problems for nonlinear higher-order PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35G30 Boundary value problems for nonlinear higher-order PDEs
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