×

The accuracy and stability of an implicit solution method for the fractional diffusion equation. (English) Zbl 1072.65123

Summary: We have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation. This model equation governs the evolution for the probability density function that describes anomalously diffusing particles. Anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur. The implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement. The accuracy of the scheme is \(O(\Delta x^2)\) in the spatial grid size and \(O(\Delta t^{1 + \gamma})\) in the fractional time step, where \(0\leqslant 1 - \gamma < 1\) is the order of the fractional derivative and \(\gamma = 1\) is standard diffusion. We have provided algebraic and numerical evidence that the scheme is unconditionally stable for \(0 < \gamma \leqslant 1\).

MSC:

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
35K05 Heat equation

Software:

FracPECE
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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
[2] Sokolov, I.; Klafter, J.; Blumen, A., Fractional kinetics, Phys. Today, 55, 48-58 (2002)
[3] Barkai, E.; Metzler, R.; Klafter, J., From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61, 132-138 (2000)
[4] Henry, B.; Wearne, S., Fractional reaction-diffusion, Physica A, 276, 448-455 (2000)
[5] Henry, B.; Wearne, S., Existence of turing instabilities in a two-species fractional reaction-diffusion system, SIAM J. Appl. Math., 62, 3, 870-887 (2002) · Zbl 1103.35047
[6] Ghosh, R.; Webb, W., Automated detection and tracking of individual and clustered cell low density lipoprotein receptor molecules, Biophys. J., 68, 766-778 (1994)
[7] Feder, T.; Brust-Mascher, I.; Slattery, J.; Baird, B.; Webb, W., Constrained diffusion or immobile fraction on cell surfaces: a new interpretation, Biophys. J., 70, 2767-2773 (1996)
[8] Sheets, E.; Lee, G.; Simson, R.; Jacobson, K., Transient confinement of a glycosylphosphatidylinositol-anchored protein in the plasma membrane, Biochemistry, 36, 12449-12458 (1997)
[9] Smith, P.; Morrison, I.; Wilson, K.; Fernandez, N.; Cherry, R., Anomalous diffusion of major histocompatability complex class I molecules on HeLa cells determined by single particle tracking, Biophys. J., 76, 3331-3344 (1999)
[10] Brown, E.; Wu, E.; Zipfel, W.; Webb, W., Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery, Biophys. J., 77, 2837-2849 (1999)
[12] Diethelm, K.; Walz, G., Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms, 16, 231-253 (1997) · Zbl 0926.65070
[14] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal., 5, 1-6 (1997) · Zbl 0890.65071
[16] Diethelm, K.; Ford, N.; Freed, A., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlin. Dynamics, 29, 3-22 (2002) · Zbl 1009.65049
[18] Podlubny, I., Numerical solution of ordinary fractional differential equations by the fractional difference method, (Proceedings of the Second International Conference in Difference Equations (1997), Gordon and Breach: Gordon and Breach London), 507-515 · Zbl 0893.65051
[20] Oldham, K.; Spanier, J., The fractional calculus: theory and applications of differentiation and integration to arbitrary order, Mathematics in Science and Engineering, vol. 111 (1974), Academic Press: Academic Press New York and London · Zbl 0292.26011
[22] Podlubny, I., Fractional differential equations, Mathematics in Science and Engineering, vol. 198 (1999), Academic Press: Academic Press New York and London · Zbl 0918.34010
[23] Lynch, V.; Carreras, B.; del-Castillo-Negrete, D.; Ferreira-Mejias, K.; Hicks, H., Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys., 192, 406-421 (2003) · Zbl 1047.76075
[24] Sanz-Serna, J., A numerical method for a partial integro-differential equation, SIAM J. Numer. Anal., 25, 2, 319-327 (1988) · Zbl 0643.65098
[25] López-Marcos, J., A difference scheme for a nonlinear partial integrodifferential equation, SIAM J. Numer. Anal., 27, 1, 20-31 (1990) · Zbl 0693.65097
[26] Adolfsson, K.; Enelund, M.; Larsson, S., Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel, Comp. Meth. Appl. Mech. Eng., 192, 5285-5304 (2003) · Zbl 1042.65103
[27] Lubich, C.; Sloan, I.; Thomée, V., Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comput., 65, 213, 1-17 (1996) · Zbl 0852.65138
[28] McLean, W.; Thomée, V., Numerical solution of an evolution equation with a positive-type memory term, J. Aust. Math. Soc. Ser. B, 35, 23-70 (1993) · Zbl 0791.65105
[29] McLean, W.; Thomée, V.; Wahlbin, L., Discretization with variable time steps of an evolution equation with a positive-type memory term, J. Comput. Appl. Math., 69, 49-69 (1996) · Zbl 0858.65143
[30] Lubich, C., Discretized fractional calculus, SIAM J. Math. Anal., 17, 3, 704-719 (1986) · Zbl 0624.65015
[31] McLean, W.; Thomée, V., Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal., 24, 3, 439-463 (2004) · Zbl 1068.65146
[32] Henry, B.; Langlands, T.; Wearne, S., Spatiotemporal patterning and turbulence in fractional activator-inhibitor systems, (Mehauté, A. L.; Machado, J. T.; Trigeassou, J.; Sabatier, J., Proceedings of the First International Workskshop on Fractional Differentiation and it’s Applications (2004), International Federation of Automatic Control: International Federation of Automatic Control Bordeaux, France), 113-120
[33] Metzler, R.; Klafter, J., Boundary value problems for fractional diffusion equations, Physica A, 278, 107-125 (2000)
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.