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Finite difference methods for fractional dispersion equations. (English) Zbl 1193.65158
Summary: The fractional weighted average finite difference method for space-fractional advection-dispersion equation is proposed, which is based on shifted Grünwald formula. This method is unconditionally stable, consistent and convergent. A numerical example is given, and the numerical results verify the theoretical conclusions.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35R11 Fractional partial differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Sousa, E., Finite difference approximates for a fractional advection diffusion problem, J. comput. phys., 228, 4038-4054, (2009) · Zbl 1169.65126
[2] Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K., Stability and convergence of the difference methods for the space-time fractional advection – diffusion equation, Appl. math. comput., 191, 2-20, (2007) · Zbl 1193.76093
[3] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. numer. anal., 46, 1079-1095, (2008) · Zbl 1173.26006
[4] Raberto, M.; Scalas, E.; Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A, 314, 749-755, (2002) · Zbl 1001.91033
[5] Sabatelli, L.; Keating, S.; Dudley, J.; Richmond, P., Waiting time distributions in financial markets, Eur. phys. J. B, 27, 273-275, (2002)
[6] Galue, L.; Kalla, S.L.; Al-Saqabi, B.N., Fractional extensions of the temperature field problems in oil strata, Appl. math. comput., 186, 35-44, (2007) · Zbl 1110.76050
[7] Li, X.; Xu, M.; Jiang, X., Homotopy perturbation method to time-fractional diffusion equation with a moving boundary, Appl. math. comput., 208, 434-439, (2009) · Zbl 1159.65106
[8] Odibat, Z.; Momani, S.; Erturk, V.S., Generalized differential transform method: application to differential equations of fractional order, Appl. math. comput., 197, 67-477, (2008) · Zbl 1141.65092
[9] Fix, G.J.; Roop, J.P., Least squares finite-element solution of a fractional order two point boundary value problem, Comput. math. appl., 48, 1017-1033, (2004) · Zbl 1069.65094
[10] Metzler, R.; Klafter, J., The random walks guide to anomalous diffusion: a fractional dynamics approach, Phys. rep., 339, 1-77, (2000) · Zbl 0984.82032
[11] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[12] Deng, W., Finite element method for the space and time fractional fokker – planck equation, SIAM J. numer. anal., 47, 204-226, (2008) · Zbl 1416.65344
[13] Yuste, S.B., Weighted average finite difference methods for fractional diffusion equations, J. comput. phys., 216, 264-274, (2006) · Zbl 1094.65085
[14] Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional fokker – planck equation, J. comput. appl. math., 166, 209-219, (2004) · Zbl 1036.82019
[15] Meerschaert, M.M.; Tadjeran, C., Finite difference approximations for fractional advection – dispersion flow equations, J. comput. appl. math., 172, 65-77, (2004) · Zbl 1126.76346
[16] Morton, K.W.; Mayers, D.F., Numerical solution of partial differential equations, (1994), Cambridge University Press Cambridge · Zbl 0811.65063
[17] Samko, S.; Kilbas, A.; Marichev, O., Fractional intergrals and derivatives: theory and applications, (1993), Gordon and Breach London · Zbl 0818.26003
[18] Tadjeran, C.; Meerschaert, M.M., A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. comput. phys., 220, 13-823, (2007) · Zbl 1113.65124
[19] Isaacson, E.; Keller, H.B., Analysis of numerical methods, (1966), Wiley New York · Zbl 0168.13101
[20] Su, L.; Wang, W.; Yang, Z., Finite difference approximations for the fractional advection – diffusion equation, Phys. lett. A, 373, 4405-4408, (2009) · Zbl 1234.65034
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