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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.
MSC:
65R20Integral equations (numerical methods)
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
26A33Fractional derivatives and integrals (real functions)
45K05Integro-partial differential equations
26A33Fractional derivatives and integrals (real functions)
References:
[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 · doi:10.1016/j.cam.2004.01.033
[2]Podlubny, I.: Fractional differential equations, (1999)
[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 · doi:10.1016/j.jcp.2005.05.017
[4]Duan, J. S.: Time- and space-fractional partial differential equations, J. math. Phys. 46, 1063-1071 (2005) · Zbl 1076.26006 · doi:10.1063/1.1819524
[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 · doi:10.1016/j.jcp.2004.11.025
[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)