×

Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term. (English) Zbl 1203.82068

Summary: Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPENST), which involve the Caputo time fractional derivative (CTFD) of order \(\alpha \in (0, 1)\) and the symmetric Riesz space fractional derivative (RSFD) of order \(\mu \in (1, 2]\). Approximating the CTFD and RSFD using the L1-algorithm and shifted Grünwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

MSC:

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
65C30 Numerical solutions to stochastic differential and integral equations
26A33 Fractional derivatives and integrals
35Q84 Fokker-Planck equations

Software:

FODE
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, vol. 18 of Springer Series in Synergetics, Springer, Berlin, Germany, 2nd edition, 1989. · Zbl 0665.60084
[2] D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation,” Water Resources Research, vol. 36, no. 6, pp. 1403-1412, 2000. · doi:10.1029/2000WR900031
[3] D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “The fractional-order governing equation of Lévy motion,” Water Resources Research, vol. 36, no. 6, pp. 1413-1423, 2000. · doi:10.1029/2000WR900032
[4] J.-P. Bouchaud and A. Georges, “Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,” Physics Reports, vol. 195, no. 4-5, pp. 127-293, 1990. · doi:10.1016/0370-1573(90)90099-N
[5] R. Metzler and J. Klafter, “Fractional Fokker-Planck equation: dispersive transport in an external force field,” Journal of Molecular Liquids, vol. 86, no. 1, pp. 219-228, 2000. · doi:10.1016/S0167-7322(99)00143-9
[6] G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,” Physics Reports, vol. 371, no. 6, pp. 461-580, 2002. · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9
[7] F. Liu, V. Anh, and I. Turner, “Numerical solution of the space fractional Fokker-Planck equation,” Journal of Computational and Applied Mathematics, vol. 166, no. 1, pp. 209-219, 2004. · Zbl 1036.82019 · doi:10.1016/j.cam.2003.09.028
[8] R. Gorenflo and F. Mainardi, “Random walk models for space-fractional diffusion processes,” Fractional Calculus & Applied Analysis, vol. 1, no. 2, pp. 167-191, 1998. · Zbl 0946.60039
[9] C. W. Chow and K. L. Liu, “Fokker-Planck equation and subdiffusive fractional Fokker-Planck equation of bistable systems with sinks,” Physica A, vol. 341, no. 1-4, pp. 87-106, 2004. · doi:10.1016/j.physa.2004.04.114
[10] S. B. Yuste and L. Acedo, “An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations,” SIAM Journal on Numerical Analysis, vol. 42, no. 5, pp. 1862-1874, 2005. · Zbl 1119.65379 · doi:10.1137/030602666
[11] T. A. M. Langlands and B. I. Henry, “The accuracy and stability of an implicit solution method for the fractional diffusion equation,” Journal of Computational Physics, vol. 205, no. 2, pp. 719-736, 2005. · Zbl 1072.65123 · doi:10.1016/j.jcp.2004.11.025
[12] C.-M. Chen, F. Liu, I. Turner, and V. Anh, “A Fourier method for the fractional diffusion equation describing sub-diffusion,” Journal of Computational Physics, vol. 227, no. 2, pp. 886-897, 2007. · Zbl 1165.65053 · doi:10.1016/j.jcp.2007.05.012
[13] P. Zhuang, F. Liu, V. Anh, and I. Turner, “New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation,” SIAM Journal on Numerical Analysis, vol. 46, no. 2, pp. 1079-1095, 2008. · Zbl 1173.26006 · doi:10.1137/060673114
[14] C.-M. Chen, F. Liu, and V. Anh, “A Fourier method and an extrapolation technique for Stokes/ first problem for a heated generalized second grade fluid with fractional derivative,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 777-789, 2009. · Zbl 1153.76049 · doi:10.1016/j.cam.2008.03.001
[15] S. Chen, F. Liu, P. Zhuang, and V. Anh, “Finite difference approximations for the fractional Fokker-Planck equation,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 256-273, 2009. · Zbl 1167.65419 · doi:10.1016/j.apm.2007.11.005
[16] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999. · Zbl 0924.34008
[17] A. Schot, M. K. Lenzi, L. R. Evangelista, L. C. Malacarne, R. S. Mendes, and E. K. Lenzi, “Fractional diffusion equation with an absorbent term and a linear external force: exact solution,” Physics Letters A, vol. 366, no. 4-5, pp. 346-350, 2007. · doi:10.1016/j.physleta.2007.02.056
[18] E. K. Lenzi, L. C. Malacarne, R. S. Mendes, and I. T. Pedron, “Anomalous diffusion, nonlinear fractional Fokker-Planck equation and solutions,” Physica A, vol. 319, pp. 245-252, 2003. · Zbl 1008.82027 · doi:10.1016/S0378-4371(02)01495-4
[19] P. Zhuang, F. Liu, V. Anh, and I. Turner, “Numerical treatment for the fractional Fokker-Planck equation,” The ANZIAM Journal, vol. 48, pp. C759-C774, 2006-2007. · Zbl 1334.82048
[20] B. Baeumer, M. Kovács, and M. M. Meerschaert, “Numerical solutions for fractional reaction-diffusion equations,” Computers & Mathematics with Applications, vol. 55, no. 10, pp. 2212-2226, 2008. · Zbl 1142.65422 · doi:10.1016/j.camwa.2007.11.012
[21] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 11 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011
[22] M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for two-sided space-fractional partial differential equations,” Applied Numerical Mathematics, vol. 56, no. 1, pp. 80-90, 2006. · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008
[23] R. Lin and F. Liu, “Fractional high order methods for the nonlinear fractional ordinary differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 4, pp. 856-869, 2007. · Zbl 1118.65079 · doi:10.1016/j.na.2005.12.027
[24] F. Liu, C. Yang, and K. Burrage, “Numerical method and analytic technique of the modified anomalous subdiffusion equation with a nonlinear source term,” Journal of Computational and Applied Mathematics, vol. 231, no. 1, pp. 160-176, 2009. · Zbl 1170.65107 · doi:10.1016/j.cam.2009.02.013
[25] M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for fractional advection-dispersion flow equations,” Journal of Computational and Applied Mathematics, vol. 172, no. 1, pp. 65-77, 2004. · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[26] W. Deng, “Finite element method for the space and time fractional Fokker-Planck equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 204-226, 2008-2009. · Zbl 1416.65344 · doi:10.1137/080714130
[27] J. I. Ramos, “Damping characteristics of finite difference methods for one-dimensional reaction-diffusion equations,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 607-609, 2006. · Zbl 1107.65076 · doi:10.1016/j.amc.2006.04.023
[28] D. del-Castillo-Negrete, B. A. Carreras, and V. E. Lynch, “Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach,” Physical Review Letters, vol. 91, no. 1, Article ID 018302, 4 pages, 2003. · doi:10.1103/PhysRevLett.91.018302
[29] V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, and H. R. Hicks, “Numerical methods for the solution of partial differential equations of fractional order,” Journal of Computational Physics, vol. 192, no. 2, pp. 406-421, 2003. · Zbl 1047.76075 · doi:10.1016/j.jcp.2003.07.008
[30] B. Baeumer, M. Kovács, and M. M. Meerschaert, “Fractional reproduction-dispersal equations and heavy tail dispersal kernels,” Bulletin of Mathematical Biology, vol. 69, no. 7, pp. 2281-2297, 2007. · Zbl 1296.92195 · doi:10.1007/s11538-007-9220-2
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.