zbMATH — the first resource for mathematics

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. (English) Zbl 1185.65200
Summary: We consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection-dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order \(\alpha\in (1,2]\). The RFADE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order \(\beta \in (0,1)\) and of order \(\alpha \in (1,2]\), respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
Full Text: DOI
[1] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley · Zbl 0789.26002
[2] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[3] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[4] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Amsterdam · Zbl 0818.26003
[5] Yuste, S.B.; Lindenberg, K., Subdiffusion-limited A+A reactions, Phys. rev. lett., 87, 11, 118301, (2001)
[6] Barkai, E.; Metzler, R.; Klafter, J., From continuous time random walks to the fractional fokker – planck equation, Phys. rev. E, 61, 1, 132-138, (2000)
[7] Metzler, R.; Klafter, J., Boundary value problems for fractional diffusion equations, Physica A: stat. mech. appl., 278, 1-2, 107-125, (2000)
[8] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. rep., 339, 1, 1-77, (2000) · Zbl 0984.82032
[9] Saichev, A.I.; Zaslavsky, G.M., Fractional kinetic equations: solutions and applications, Chaos, 7, 4, 753-764, (1997) · Zbl 0933.37029
[10] Zaslavsky, G.M., Chaos, fractional kinetics, and anomalous transport, Phys. rep., 371, 6, 461-580, (2002) · Zbl 0999.82053
[11] Yuste, S.B.; Acedo, L.; Lindenberg, K., Reaction front in an A+B→C reaction – subdiffusion process, Phys. rev. E, 69, 3, 036126, (2004)
[12] Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., Application of a fractional advection – dispersion equation, Water resour. res., 36, 6, 1403-1412, (2000)
[13] Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., The fractional-order governing equation of Lévy motion, Water resour. res., 36, 6, 1413-1423, (2000)
[14] Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional fokker – planck equation, J. comput. appl. math., 166, 1, 209-219, (2004) · Zbl 1036.82019
[15] Liu, F.; Anh, V.; Turner, I.; Zhuang, P., Time fractional advection – dispersion equation, J. appl. math. comput., 13, 1-2, 233-246, (2003) · Zbl 1068.26006
[16] Gorenflo, R.; Mainardi, F.; Scalas, E.; Raberto, M., Fractional calculus and continuous-time finance. III. the diffusion limit, Math. finance (Konstanz, 2000), 171-180, (2001) · Zbl 1138.91444
[17] Raberto, M.; Scalas, E.; Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Phys. A: stat. mech. appl., 314, 1-4, 749-755, (2002) · Zbl 1001.91033
[18] Scalas, E.; Gorenflo, R.; Mainardi, F., Fractional calculus and continuous-time finance, Phys. A: stat. mech. appl., 284, 1-4, 376-384, (2000)
[19] Wyss, W., The fractional black – scholes equation, Fract. calculus appl. anal., 3, 51-61, (2000) · Zbl 1058.91045
[20] Momani, S.; Odibat, Z., Numerical solutions of the space – time fractional advection – dispersion equation, Numer. meth. partial differ. equat., 24, 6, 1416-1429, (2008) · Zbl 1148.76044
[21] Hilfer, R., Fractional diffusion based on riemann – liouville fractional derivatives, J. phys. chem. B, 104, 3914-3917, (2000)
[22] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space – time fractional diffusion equation, Fract. calculus appl. anal., 4, 2, 153-192, (2001) · Zbl 1054.35156
[23] Huang, F.; Liu, F., The fundamental solution of the space – time fractional advection – dispersion equation, J. appl. math. comput., 18, 1-2, 339-350, (2005) · Zbl 1086.35003
[24] 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
[25] Liu, Q.; Liu, F.; Turner, I.; Anh, V., Approximation of the Lévy – feller advection – dispersion process by random walk and finite difference method, J. phys. comp., 222, 1, 57-70, (2007) · Zbl 1112.65006
[26] M. Ciesielski, J. Leszczynski, Numerical solutions of a boundary value problem for the anomalous diffusion equation with the Riesz fractional derivative, in: Proceedings of the 16th International Conference on Computer Methods in Mechanics Czestochowa, Poland, 2005.
[27] Meerschaert, M.M.; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. numer. math., 56, 1, 80-90, (2006) · Zbl 1086.65087
[28] Shen, S.; Liu, F.; Anh, V.; Turner, I., The fundamental solution and numerical solution of the Riesz fractional advection – dispersion equation, IMA J. appl. math., 73, 6, 850-872, (2008) · Zbl 1179.37073
[29] Meerschaert, M.M.; Scheffler, H.P.; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. comput. phys., 211, 1, 249-261, (2006) · Zbl 1085.65080
[30] Gorenflo, R.; Mainardi, F., Random walk models for space-fractional diffusion processes, Fract. calculus appl. anal., 1, 2, 167-191, (1998) · Zbl 0946.60039
[31] Ilić, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation, Fract. calculus appl. anal., 8, 3, 323-341, (2005) · Zbl 1126.26009
[32] Brenan, K.E.; Campbell, S.L.; Petzold, L.R., Numerical solution of initial-value problems in differential-algebraic equations, (1989), North-Holland New York · Zbl 0699.65057
[33] Ilić, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation (II)—with nonhomogeneous boundary conditions, Fract. calculus appl. anal., 9, 4, 333-349, (2006) · Zbl 1132.35507
[34] Gear, C.W., The automatic integration of ordinary differential equations, Comm. ACM, 4, 3, 176-179, (1971) · Zbl 0217.21701
[35] F. Liu, Numerical analysis of semiconductor device equations in two and three dimensions, Ph.D. Thesis, Trinity College, Dublin, Ireland, 1991.
[36] Tocci, M.D.; Kelley, C.T., Accurate and economical solution of the pressure-head from of richards’ equation by the method of lines, Water resource, 20, 1, 1-14, (1997)
[37] Bhatia, S.K.; Liu, F.; Arvind, G., Effect of pore blockage on adsorption isotherms and dynamics: anomalous adsorption of iodine on activated carbon, Langmuir, 16, 8, 4001-4008, (2000)
[38] Liu, F.; Bhatia, S.K., Computationally efficient solution techniques for adsorption problems involving steep gradients in bidisperse particles, Comp. chem. eng., 23, 933-943, (1999)
[39] Liu, F.; Bhatia, S.K., Solution techniques for transport problems involving steep concentration gradients: application to noncatalytic fluid solid reactions, Comp. chem. eng., 25, 9-10, 1159-1168, (2001)
[40] Liu, F.; Bhatia, S.K.; Abarzhi, I.I., Numerical solution of hyperbolic models of transport in bidisperse solids, Comp. chem. eng., 24, 8, 1981-1995, (2000)
[41] Liu, F.; Bhatia, S.K., Application of petrov – galerkin methods to transient boundary value problems in chemical engineering: adsorption with steep gradients in bidisperse solids, Chem. eng. sci., 56, 12, 3727-3735, (2001)
[42] Liu, F.; Turner, I.; Anh, V., An unstructured mesh finite volume method for modelling saltwater intrusion into coastal aquifers, Korean J. comput. appl. math., 9, 2, 391-407, (2002) · Zbl 1002.76074
[43] Liu, F.; Turner, I.; Anh, V.; Su, N., A two-dimensional finite volume method for transient simulation of time-, scale- and density-dependent transport in heterogeneous aquifer systems, J. appl. math. comp., 11, 215-241, (2003) · Zbl 1145.76407
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.