Finite difference/spectral approximations for the time-fractional diffusion equation. (English) Zbl 1126.65121

Consider the problem of finding a numerical solution for the time-fractional diffusion equation with the fractional derivative with respect to time being given in the sense of Caputo. For the solution of this problem, the authors propose a method based on a finite difference scheme with respect to time combined with a Legendre spectral method for the space variable(s). A proof of stability and convergence of the algorithm is provided. Error estimates and numerical results are given as well.


65R20 Numerical methods for integral equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
45K05 Integro-partial differential equations
26A33 Fractional derivatives and integrals
35K15 Initial value problems for second-order parabolic equations
Full Text: DOI


[1] Agrawal, O. P., Solution for a fractional diffusion-wave equation defined in a bounded domain, J. Nonlinear Dynam., 29, 145-155 (2002) · Zbl 1009.65085
[2] Berberan-Santos, M. N., Properties of the Mittag-Leffler relaxation function, J. Math. Chem., 38, 4, 629-635 (2005) · Zbl 1101.33015
[3] Bernardi, C.; Maday, Y., Approximations Spectrales de Problems aux Limites Elliptiques (1992), Springer-Verlag: Springer-Verlag Berlin · Zbl 0773.47032
[4] Diethelm, K.; Ford, N. J., Numerical solution of the Bagley-Torvik equation, BIT, 42, 490-507 (2002) · Zbl 1035.65067
[5] Diethelm, K.; Freed, A. D., On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, (Proceedings of the Second Conference on Scientific Computing in Chemical Engineering (1999), Springer: Springer Heidelberg), 217-224
[6] Deng, Z.; Singh, V. P.; Bengtsson, L., Numerical solution of fractional advection-dispersion equation, J. Hydraulic Eng., 130, 422-431 (2004)
[7] 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
[8] Gorenflo, R.; Luchko, Y.; Mainardi, F., Wright function as scale-invariant solutions of the diffusion-wave equation, J. Comp. Appl. Math., 118, 175-191 (2000) · Zbl 0973.35012
[9] Gorenflo, R.; Mainardi, F.; Moretti, D.; Paradisi, P., Time fractional diffusion: a discrete random walk approach, Nonlinear Dynam., 29, 129-143 (2002) · Zbl 1009.82016
[10] Huang, F.; Liu, F., The time fractional diffusion and advection-dispersion equation, ANZIAM J., 46, 1-14 (2005) · Zbl 1072.35218
[11] Liu, F.; Anh, V.; Turner, I.; Zhuang, P., Time fractional advection dispersion equation, J. Appl. Math. Comput., 13, 233-245 (2003) · Zbl 1068.26006
[12] Liu, F.; Shen, S.; Anh, V.; Turner, I., Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J., 46, E, 488-504 (2005) · Zbl 1082.60511
[13] Lynch, V. E.; Carreras, B. A.; Del-Castillo-Negrete, D.; Ferreira-Mejias, K. M.; Hicks, H. R., Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys., 192, 406-421 (2003) · Zbl 1047.76075
[14] Mainardi, F., Some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer: Springer Wien), 291-348 · Zbl 0917.73004
[15] Meerschaert, M.; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comp. Appl. Math., 172, 65-77 (2004) · Zbl 1126.76346
[16] Podlubny, I., Fractional Differential Equations (1999), Academic Press · Zbl 0918.34010
[17] Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations (1994), Springer-Verlag: Springer-Verlag Berlin · Zbl 0852.76051
[18] Schneider, W. R.; Wyss, W., Fractional diffusion and wave equations, J. Math. Phys., 30, 34-144 (1989) · Zbl 0692.45004
[19] Tang, T., A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math., 11, 4, 309-319 (1993) · Zbl 0768.65093
[20] Tadjeran, C.; Meerschaert, M.; Scheffler, H. P., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213, 205-213 (2006) · Zbl 1089.65089
[21] Wyss, W., The fractional diffusion equation, J. Math. Phys., 27, 2782-2785 (1986) · Zbl 0632.35031
[22] Xu, C. J.; Lin, Y. M., Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation, Inter. J. Numer. Met. Fluids, 32, 619-646 (2000) · Zbl 0981.76066
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.