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Finite difference approximations for two-sided space-fractional partial differential equations. (English) Zbl 1086.65087

The authors examine some practical numerical finite difference methods for solving initial-boundary value problems for fractional order partial differential equations. Such equations are generalizations of classical partial differential equations and used to describe fluid flow and finance flow models. The case when left-handed or right-handed fractional spatial derivative may be present in the partial differential equation is considered.

Stability, consistency and therefore also convergence of the proposed method are discussed. The convergence and stability results unify the corresponding results for classical parabolic and hyperbolic partial differential equations into a single condition. A numerical example using finite difference methods for two-sides fractional equations are presented and compared with the exact analytical solutions.

MSC:
65M06Finite difference methods (IVP of PDE)
26A33Fractional derivatives and integrals (real functions)
35R05PDEs with discontinuous coefficients or data
65M12Stability and convergence of numerical methods (IVP of PDE)