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Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. (English) Zbl 1193.76093

76M20Finite difference methods (fluid mechanics)
65M06Finite difference methods (IVP of PDE)
Full Text: DOI
[1] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M.: Application of a fractional advection -- dispersion equation. Water resour. Res. 36, No. 6, 1403-1412 (2000)
[2] J.S. Ervin, J.P. Roop, Variational solution of fractional advection dispersion equations on bounded domains in Rd, in press. · Zbl 1117.65169
[3] Huang, F.; Liu, F.: The fundamental solution of the space -- time fractional advection -- dispersion equation. J. appl. Math. comput. 19, 233-245 (2005)
[4] Liu, F.; Anh, V.; Turner, I.; Zhuang, P.: Time fractional advection dispersion equation. J. appl. Math. comput. 13, 233-245 (2003) · Zbl 1068.26006
[5] Liu, F.; Anh, V.; Turner, I.: Numerical solution of space fractional Fokker -- Planck equation. J. comput. Appl. math. 166, 209-219 (2004) · Zbl 1036.82019
[6] Q. Liu, F. Liu, I. Turner, V. Anh, Approximation of the Levy -- Feller advection -- dispersion process by random walk and finite difference method, J. Phys. Comput., in press. · Zbl 1112.65006
[7] Meerschaert, M.; Tadjeran, C.: Finite difference approximations for fractional advection -- dispersion flow equations. J. comput. Appl. math. 172, 65-77 (2004) · Zbl 1126.76346
[8] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. rep. 339, 1-77 (2000) · Zbl 0984.82032
[9] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations. (1993) · Zbl 0789.26002
[10] S. Momani, Z. Odibat, Numerical solutions of the space -- time fractional advection -- dispersion equation, J. Appl. Math. Comput., in press. · Zbl 1148.76044
[11] Oldham, K. B.; Spanier, J.: The fractional calculus. (1974) · Zbl 0292.26011
[12] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[13] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications. (1993) · Zbl 0818.26003
[14] P. Zhuang, F. Liu, Implicit difference approximation for the time fractional diffusion equation, J. Appl. Math. Comput., in press. · Zbl 1140.65094