zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Finite difference approximations for the fractional Fokker-Planck equation. (English) Zbl 1167.65419
Summary: The fractional Fokker-Planck equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine some practical numerical methods to solve a class of initial-boundary value problems for the fractional Fokker-Planck equation on a finite domain. The solvability, stability, consistency, and convergence of these methods are discussed. Their stability is proved by the energy method. Two numerical examples are also presented to evaluate these finite difference methods against the exact analytical solutions.
MSC:
65M06Finite difference methods (IVP of PDE)
26A33Fractional derivatives and integrals (real functions)
35K55Nonlinear parabolic equations
References:
[1]Risken, H.: The Fokker – Planck equation, (1989)
[2]Metzler, R.; Barkai, E.; Klafter, J.: Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker – Planck equation approach, Phys. rev. Lett. 82, 3563-3567 (1999)
[3]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 · doi:10.1016/S0370-1573(00)00070-3
[4]Metzler, R.; Klafter, J.: The fractional Fokker – Planck equation, dispersive transport in an external field, J. mol. Liq. 86, 219-228 (2000)
[5]Bouchaud, J. P.; Georges, A.: Anomalous diffusion in disordered media, Phys. rep. 195, 127-293 (1990)
[6]Scher, H.; Shlesinger, M. F.; Bendler, J. T.: Time-scale invariance in transport and relaxation, Phys. today 44, 26-34 (1991)
[7]Klemm, A.; Müller, H. P.; Kimmich, R.: Evaluation of fractal parameters of percolation model objects and natural porous media by means of NMR microscopy, Physica A. 266, 242-246 (1999)
[8]Amblard, F.; Maggs, A. C.; Yurke, B.; Pargellis, A. N.; Leibler, S.: Subdiffusion and anomalous local viscoelasticity in actin networks, Phys. rev. Lett. 77, 4470-4473 (1996)
[9]Pfister, G.; Scher, H.: Dispersive (non-Gaussian) transient transport in disordered solids, Adv. phys. 27, 747-798 (1978)
[10]Gu, Q.; Schiff, E. A.; Grebner, S.; Wang, F.; Schwarz, R.: Non-Gaussian transport measurements and the Einstein relation in amorphous silicon, Phys. rev. Lett. 76, 3196-3199 (1996)
[11]Metzler, R.; Klafter, J.: Lévy meets Boltzmann: strange initial conditions for Brownian and fractional Fokker – Planck equations, Physica A 302, 290-296 (2001) · Zbl 0979.82050 · doi:10.1016/S0378-4371(01)00472-1
[12]Sokolov, I. M.; Blumen, A.; Klafter, J.: Linear response in complex systems: CTRW and the fractional Fokker – Planck equations, Physica A 302, 268-278 (2001) · Zbl 0983.60040 · doi:10.1016/S0378-4371(01)00470-8
[13]So, F.; Liu, K. L.: A study of the subdiffusive fractional Fokker – Planck equation of bistable systems, Physica A 331, 378-390 (2004)
[14]Hilfe, R.: Applications of fractional calculus in physics, (1999)
[15]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[16]Oldham, K. B.; Spanier, J.: The fractional calculus, (1974)
[17]Ross, B.: Fractional calculus and its applications, Lecture notes in mathematics 457 (1975) · Zbl 0293.00010
[18]Yuste, S. B.; Acedo, L.: An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. anal. 42, No. 5, 1862-1874 (2005) · Zbl 1119.65379 · doi:10.1137/030602666
[19]Langlands, T. A. M.; Henry, B. I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. comput. Phys. 205, 719-736 (2005) · Zbl 1072.65123 · doi:10.1016/j.jcp.2004.11.025
[20]Zhuang, P.; Liu, F.; Anh, V.; Turner, I.: New solution and analytical techniques of the implicit numerical method for the anomalous sub-diffusion equation, SIAM J. Numer. anal. (2008)
[21]Chen, C.; Liu, F.; Turner, I.; Anh, V.: A Fourier method for the fractional diffusion equation describing sub-diffusion, J. comput. Phys. 227, 886-897 (2007) · Zbl 1165.65053 · doi:10.1016/j.jcp.2007.05.012
[22]Liu, F.; Anh, V.; Turner, I.: Numerical solution of space fractional Fokker – Planck equation, J. comput. Appl. math. 166, 209-219 (2004) · Zbl 1036.82019 · doi:10.1016/j.cam.2003.09.028
[23]Liu, F.; Anh, V.; Turner, I.: Numerical simulation for solute transport in fractal porous media, Anziam j. (E) 45, 461-473 (2004)
[24]Heinsalu, E.; Patriarca, M.; Goychuk, I.; Schmid, G.; Hänggi, P.: Fractional Fokker – Planck dynamics: numerical algorithm and simulations, Phys. rev. E 73, No. 046133, 1-9 (2006)
[25]Podlubny, I.: Fractional differential equations, (1999)
[26]Hu, J.; Tang, H.: Numerical methods for differential equations, (1999)