International Journal of Differential Equations

Stability and Convergence of an Effective Numerical Method for the Time-Space Fractional Fokker-Planck Equation with a Nonlinear Source Term

Abstract

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.

Article information

Source
Int. J. Differ. Equ., Volume 2010, Special Issue (2010), Article ID 464321, 22 pages.

Dates
Revised: 20 August 2009
Accepted: 28 September 2009
First available in Project Euclid: 26 January 2017

https://projecteuclid.org/euclid.ijde/1485399866

Digital Object Identifier
doi:10.1155/2010/464321

Mathematical Reviews number (MathSciNet)
MR2575294

Zentralblatt MATH identifier
1203.82068

Citation

Yang, Qianqian; Liu, Fawang; Turner, Ian. Stability and Convergence of an Effective Numerical Method for the Time-Space Fractional Fokker-Planck Equation with a Nonlinear Source Term. Int. J. Differ. Equ. 2010, Special Issue (2010), Article ID 464321, 22 pages. doi:10.1155/2010/464321. https://projecteuclid.org/euclid.ijde/1485399866

References

• H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, vol. 18 of Springer Series in Synergetics, Springer, Berlin, Germany, 2nd edition, 1989.MR987631 \setlengthemsep0.75pt
• 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.
• 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.
• 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.MR1081295
• 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.
• G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,” Physics Reports, vol. 371, no. 6, pp. 461–580, 2002.MR1937584
• 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.MR2057973
• 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.MR1656314
• 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.MR2092678
• 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.MR2139227
• 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.MR2135000
• 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.MR2442379
• 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.MR2383224
• 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.MR2478879
• 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.MR2458510
• 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.MR1658022
• 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.
• 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.
• 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.MR2366327
• 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.MR2413687
• 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.MR0361633
• 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.MR2186432
• 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.MR2288436
• 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.MR2532659
• 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.MR2091131
• 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.MR2452858
• 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.MR2292071
• 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.
• 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.MR2020744
• 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.MR2341872