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Finite difference approximations for the fractional advection-diffusion equation. (English) Zbl 1234.65034
Summary: Fractional order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this Letter, in order to solve the two-sided fractional advection-diffusion equation, the fractional Crank-Nicholson method (FCN) is given, which is based on shifted Grünwald-Letnikov formula. It is shown that this method is unconditionally stable, consistent and convergent. The accuracy with respect to the time step is of order \((\Delta t)^2\). A numerical example is presented to confirm the conclusions.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
60G22 Fractional processes, including fractional Brownian motion
60J60 Diffusion processes
35K57 Reaction-diffusion equations
35R11 Fractional partial differential equations
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References:
[1] Hilfer, R., Applications of fractional calculus in physics, (2000), World Scientific Singapore · Zbl 0998.26002
[2] Jafari, H.; Momani, S., Phys. lett. A, 370, 388, (2007)
[3] Chaves, A., Phys. lett. A, 239, 13, (1998)
[4] Meerschaert, M.M.; Benson, D.; Scheffler, H.P.; Baeumer, B., Phys. rev. E, 65, 1103, (2002)
[5] Liu, F.; Anh, V.; Turner, I., J. comput. appl. math., 166, 209, (2004)
[6] Meerschaert, M.M.; Tadjeran, C., Appl. numer. math., 56, 80, (2006)
[7] Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., Water resour. res., 36, 6, 1413, (2000)
[8] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[9] Mainardi, F., Chaos solitons fractals, 7, 1461, (1996)
[10] Henry, B.I.; Wearne, S.L., Physica A, 276, 448, (2000)
[11] Odibat, Z.M., Phys. lett. A, 372, 1219, (2008)
[12] Momani, S.; Odibat, Z., Phys. lett. A, 365, 345, (2007)
[13] Langlands, T.; Henry, B., J. comput. phys., 205, 2, 719, (2005)
[14] Morton, K.W.; Mayers, D.F., Numerical solution of partial differential equations, (1994), Cambridge University Press Cambridge · Zbl 0811.65063
[15] Yuste, S.B., J. comput. phys., 216, 264, (2006)
[16] Odibat, Z.M., Phys. lett. A, 370, 295, (2007)
[17] Samko, S.; Kilbas, A.; Marichev, O., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach London · Zbl 0818.26003
[18] Lubich, C., SIAM J. math. anal., 17, 704, (1986)
[19] Tadjeran, C.; Meerschaert, M.M., J. comput. phys., 220, 813, (2007)
[20] Isaacson, E.; Keller, H.B., Analysis of numerical methods, (1966), Wiley New York · Zbl 0168.13101
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