zbMATH — the first resource for mathematics

A finite difference method for fractional partial differential equation. (English) Zbl 1177.65198
The author discusses the numerical solution of some space-time fractional order partial differential equations. One practical implicit numerical method is proposed to solve a class of initial-boundary value space-time fractional convection-diffusion equations with variable coefficients. A new shifted version of the usual Grünwald finite difference approximation [see M. M. Meershaert, J. Mortensen and H. P. Scheffler, Fract. Calc. Appl. Anal. 7, No. 1, 61–81 (2004; Zbl 1084.65024)] is used for the non-local fractional derivative operator and it is proved that the method is first-order consistent and unconditionally stable, for the equation with Dirichlet boundary conditions. The convergence and error estimates of the scheme, are also discussed. One numerical example, with known exact solution, is presented.

65R20 Numerical methods for integral equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
45K05 Integro-partial differential equations
Full Text: DOI
[1] Meerschaert, M.M.; Tadjeran, C., Finite difference approximations for fractional advection – dispersion flow equations, J. comput. appl. math., 172, 65-77, (2003) · Zbl 1126.76346
[2] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[3] Meerschaert, M.M.; Scheffler, H.P.; Tadjeran, C., Finite difference method for two dimensional fractional dispersion equation, J. comput. phys., 211, 249-261, (2006) · Zbl 1085.65080
[4] Duan, J.S., Time- and space-fractional partial differential equations, J. math. phys., 46, 1063-1071, (2005)
[5] Langlands, T.A.M.; Henry, B.I., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. comput. phys., 205, 719-736, (2005) · Zbl 1072.65123
[6] Meerschaert, M.M.; Benson, D.A.; Scheffler, H.P.; Baeumer, B., Stochastic solution of space-time fractional diffusion equations, Phys. rev. E, 65, 1103-1106, (2002)
[7] Meerschaert, M.M.; Mortensen, J.; Scheffler, H.P., Vector Grünwald formula for fractional derivatives, Fract. calc. appl., 7, 61-81, (2004) · Zbl 1084.65024
[8] Meerschaert, M.M.; Benson, D.A.; Scheffler, H.P.; Kern, P.B., Governing equations and solutions of anomalous random walk limits, Phys. rev. E, 66, 102-105, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.