Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion

Journal of Computational and Applied Mathematics

Volumn 205, Issue 2, 2007, Pages 871-881

  Gorenflo, R ^a   Abdel Rehim, E A ^b  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

^b SUEZ CANAL UNIVERSITY   (Egypt)

Author keywords

Convergence in distribution; Difference schemes; Fractional derivative; Fractional diffusion; Potential well; Stochastic processes

Indexed keywords

LAPLACE EQUATION; ORDINARY DIFFERENTIAL EQUATIONS; PROBABILITY; RANDOM PROCESSES;

CONVERGENCE IN DISTRIBUTION; DIFFERENCE SCHEMES; FRACTIONAL DERIVATIVE; FRACTIONAL DIFFUSION; POTENTIAL WELL;

CONVERGENCE OF NUMERICAL METHODS;

EID: 34248159960     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2005.12.043     Document Type: Article

References (30)

1. E.A. Abdel-Rehim, Modelling and Simulating of Classical and Non-Classical Diffusion Processes by Random Walks, Mensch&Buch Verlag, ISBN 3-89820-736-6, 2004, available at 〈http://www.diss.fu-berlin.de/2004/168/index.html〉.
2. Barkai E., Metzler R., and Klafter J. From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61 1 (2000) 132-138
3. Chechkin A., Gonchar V., Klafter J., Metzler R., and Tanatarov L. Stationary states of non-linear oscillators driven by Lévy noise. Chem. Phys. 284 (2002) 233-251
4. Diethelm K., Ford N.J., Freed A.D., and Luchko Yu. Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Engng. 194 (2005) 743-773
5. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vols. 1-3, McGraw-Hill, New York, 1953-1955.
6. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, New York, London, Sydney, Toronto, 1971.
7. Gorenflo R. Fractional calculus: some numerical methods. In: Carpinteri A., and Mainardi F. (Eds). Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures vol. 378 (1997), Springer, Wien 277-290
8. R. Gorenflo, E. A. Abdel-Rehim, From power laws to fractional diffusion: the direct way, Vietnam J. Math. 32 SI (2004) 65-75.
9. Gorenflo R., and Abdel-Rehim E.A. Discrete models of time-fractional diffusion in a potential well. Fractional Calculus Appl. Anal. 8 (2005) 173-200
10. Gorenflo R., and Mainardi F. Fractional calculus: integral and differential equations of fractional order. In: Carpinteri A., and Mainardi F. (Eds). Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer, Wien, New York 223-276. http://www.fracalmo.org (Reprinted in News 010101, see 〈http://www.fracalmo.org〉)
11. Gorenflo R., and Mainardi F. Random walk models approximating symmetric space-fractional diffusion processes. In: Elschner J., Gohberg I., and Silbermann B. (Eds). Problems in Mathematical Physics (2001), Birkhäuser Verlag, Basel 120-145
12. Gorenflo R., Mainardi F., Moretti D., Pagnini G., and Paradisi P. Discrete random walk models for space-time fractional diffusion. Chem. Phys. 284 (2002) 521-541
13. Gorenflo R., Mainardi F., Moretti D., and Paradisi P. Time-fractional diffusion: a discrete random walk approach. Nonlinear Dynamics 29 (2002) 129-143
14. R. Gorenflo, S. Vessella, Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics, vol. 1461, Springer, Heidelberg, 1991.
15. Gorenflo R., and Vivoli A. Fully discrete random walks for space-time fractional diffusion equations. Signal Process. 83 (2003) 2411-2420
16. Hilfer R., and Anton L. Fractional master equations and fractal time random walks. Phys. Rev. E 51 2 (1995) R848-R851
17. R. Hilfer, R. Metzler, A. Blumen, J. Klafter (Eds.), Strange Kinetics, (Special Issue), Chemical Physics, vol. 284 Nos. (1-2), 2002.
18. Langlands T.A.M., and Henry B.I. The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205 (2005) 719-736
19. Lubich Ch. Discretized fractional calculus. SIAM J. Math. Anal. 17 (1986) 704-719
20. Lynch V.E., Carreras B.A., del-Castillo-Negrete D., Ferreira-Mejias K.M., and Hicks H.R. Numerical methods for the solution of partial differential equations of fractional order. J. Comput. Phys. 192 (2003) 406-421
21. Mainardi F., Luchko Yu., and Pagnini G. The fundamental solution of the space-time fractional diffusion equation. Fractional Calculus Appl. Anal. 4 2 (2001) 153-192. http://www.fracalmo.org [Available at www.fracalmo.org, pre-print 0101]
22. Meerschaert M.M., Scheffler H.P., and Mortensen J. Vector Grünwald formula for fractional derivatives. Fractional Calculus Appl. Anal. 7 1 (2004) 61-81
23. Meerschaert M.M., and Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 127 (2004) 65-77
24. Metzler R., and Klafter J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37 (2004) R161-R208
25. α (x). C. R. Acad. Sci. Paris 137 (1903) 554-558
26. Podlubny I. Fractional Differential Equations (1999), Academic Press, San Diego, Boston, New York, London, Sydney, Tokyo, Toronto
27. Samko S.G., Kilbas A.A., and Marichev O.I. Fractional Integrals and Derivatives (Theory and Applications) (1993), OPA, Amsterdam
28. Sokolov I.M., Klafter J., and Blumen A. Fractional kinetics. Phys. Today 55 (2002) 48-54
29. Tamarkin J.D. On integrable solutions of Abel's integral equation. Ann. Math. 2 31 (1930) 219-229
30. Yuste S.B., and Acedo L. An explicit finite difference method and a new Von Neumann type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42 (2005) 1862-1874