Discrete and continuous random walk models for space-time fractional diffusion

Nonlinear Dynamics

Volumn 38, Issue 1-4, 2004, Pages 101-116

  Gorenflo, Rudolf ^a   Vivoli, Alessandro ^b   Mainardi, Francesco ^b  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

^b UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

Anomalous diffusion; Fractional calculus; Mittag Leffler function; Power laws; Random walks; Transform methods

Indexed keywords

BROWNIAN MOVEMENT; COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; GREEN'S FUNCTION; MATHEMATICAL OPERATORS; MATHEMATICAL TECHNIQUES; MATHEMATICAL TRANSFORMATIONS;

ANOMALOUS DIFFUSION; FRACTIONAL CALCULUS; MITTAG-LEFFLER FUNCTION; POWER LAWS; RANDOM WALKS; TRANSFORM METHODS;

SPACE TIME ADAPTIVE PROCESSING;

EID: 15544368331     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-004-3749-5     Document Type: Conference Paper

References (53)

1. Chechkin, A. V., Gorenflo, R., and Sokolov, I. M., 'Retarding sub- and accelerating superdiffusion governed by distributed order fractional derivatives', Physical Review E 66, 2002, 046129/1-7.
2. Chechkin, A. V., Gorenflo, R., Sokolov, I. M., and Gonchar, V. Yu., 'Distributed order fractional diffusion equations', Fractional Calculus and Applied Analysis 6, 2003, 259-279.
3. Gorenflo, R. and Mainardi, F., 'Fractional calculus: Integral and differential equations of fractional order', inFractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri, and F. Mainardi (eds.) Springer Verlag, Vienna, 1997, pp. 223-276 [reprinted in NEWS 010101, see http://www.fracalmo.org].
4. Montroll, E. W. and Weiss, G. H., 'Random walks on lattices, II', Journal of Mathematical Physics 6, 1965, 167-181.
5. Weiss, G. H., Aspects and Applications of Random Walks, North-Holland, Amsterdam, 1994.
6. Metzler, R and Klafter, J., 'The random walk's guide to anomalous diffusion: A fractional dynamics approach', Physics Report 339, 2000, 1-77.
7. Cox, D. R., Renewal Theory, Methuen, London, 1967.
8. Kotulski, M., 'Asymptotic distributions of continuous-time random walks: A probabilistic approach', Journal of Statistical Physics 81, 1995, 777-792.
9. Baeumer, B. and Meerschaert, M. M., 'Stochastic solutions for fractional Cauchy problems', Fractional Calculus and Applied Analysis 4, 2001, 481-500.
10. Gorenflo, R. and Mainardi, F., 'Non-Markovian random walks, scaling and diffusion limits', inMini-Proceedings of the Second MaPhySto Conference on Lévy Processes: Theory and Applications, O. E. Barndorff-Nielsen (ed.), Department of Mathematics, University of Aarhus, Denmark, 21-25 January 2002 (ISSN 1398-5957), pp. 120-128, 2002 [available c/o http://www.maphysto.dk, Miscellanea no. 22].
11. Meerschaert, M. M., Benson, D. A., Scheffler, H. P., and Baeumer, B., 'Stochastic solutions of space-fractional diffusion equation', Physical Review E 65, 2002, 041103/1-4.
12. Hughes, B. D., Montroll, E. W., and Shlesinger, M. F., 'Fractal random walks', Journal of Statistical Physics 28, 1982, 111-126.
13. Gorenflo, R., Mainardi, F., Scalas, E., and Raberto, M., 'Fractional calculus and continuous-time finance III: The diffusion limit', inMathematical Finance, K. Kohlmann and S. Tang (eds.), Birkhäuser Verlag, Basel, 2001, pp. 171-180.
14. Scalas, E., Gorenflo, R., and Mainardi, F., 'Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation', Physical Review E 69, 2004, 011107/1-8.
15. Barkai, E., 'CTRW pathways to the fractional diffusion equation', Chemical Physics 284, 2002, 13-27.
16. Barkai, E., Metzler, R., and Klafter, J. 'From continuous-time random walks to the fractional Fokker-Planck equation', Physical Review E 61, 2000, 132-138.
17. Hilfer, R., 'Exact solutions for a class of fractal time random walks', Fractals 3, 1995, 211-216.
18. Hilfer, R. and Anton, L., 'Fractional integral equations and fractal time random walks', Physical Review E 51, 1995, R848-R851.
19. Meerschaert, M. M., Benson, D. A., Scheffler, H. P., and Becker-Kern, P., 'Governing equations and solutions of anomalous random walk limits', Physical Review E 66, 2002, 060102/1-4.
20. Saichev, A. I. and Zaslavsky, G. M., 'Fractional kinetic equations: Solutions and applications', Chaos 7, 1997, 753-764.
21. Gorenflo, R., Iskenderov, A., and Luchko, Yu., 'Mapping between solutions of fractional diffusion-wave equations', Fractional Calculus and Applied Analysis 3, 2000, 75-86.
22. Mainardi, F., Luchko, Yu., and Pagnini, G., 'The fundamental solution of the space-time fractional diffusion equation', Fractional Calculus and Applied Analysis 4, 2001, 153-192 [reprinted in NEWS 010401, see http://www.fracalmo.org].
23. Feller, W., An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, New York, 1971.
24. Mainardi, F., Raberto, M., Gorenflo, R., and Scalas, E., 'Fractional calculus and continuous-time finance II: The waiting-time distribution', Physica A 287, 2000, 468-481.
25. Scalas, E., Gorenflo, R., and Mainardi, F., 'Fractional calculus and continuous-time finance', Physica A 284, 2000, 376-384.
26. Gillis, J. E. and Weiss, G. H., 'Expected number of distinct sites visited by a random walk with an infinite variance', Journal of Mathematical Physics 11, 1970, 1307-1312.
27. Gorenflo, R. and Mainardi, F., 'Random walk models approximating symmetric space-fractional diffusion processes', inProblems in Mathematical Physics, J. Elschner, I. Gohberg and B. Silbermann (eds.), Birkhäuser Verlag, Basel, 2001, pp. 120-145.
28. Gorenflo, R. and Abdel-Rehim, E., 'From power laws to fractional diffusion', Vietnam Journal of Mathematics 32, 2004 (in press).
29. Gnedenko, B. V. and Kolmogorov, A. N., Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, Massachusetts, 1954.
30. Balakrishnan, V., 'Anomalous diffusion in one dimension', Physica A 132, 1985, 569-580.
31. Uchaikin, V. V. and Saenko, V. V., 'Stochastic solution of partial differential equations of fractional order', Siberian Journal of Numerical Mathematics 6, 2003, 197-203.
32. Hilfer, R., 'On fractional diffusion and continuous time random walks', Physica A 329, 2003, 35-39.
33. Gorenflo, R. and Vivoli, A., 'Fully discrete random walks for space-time fractional diffusion equations', Signal Processing 83, 2003, 2411-2420.
34. Gorenflo, R., De Fabritiis, G., and Mainardi, F., 'Discrete random walk models for symmetric Lévy-Feller diffusion processes', Physica A 269, 1999, 79-89.
35. Gorenflo, R. and Mainardi, F., 'Fractional diffusion processes: probability distributions and continuous time random walk', inProcesses with Long Range Correlations, G. Rangarajan and M. Ding (eds.), Springer Verlag, Berlin 2003, pp. 148-166 [Lecture Notes in Physics, No. 621].
36. Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., and Paradisi, P., 'Discrete random walk models for space-time fractional diffusion', Chemical Physics 284, 2002, 521-544.
37. Gorenflo, R., Mainardi, F., Moretti, D., and Paradisi, P., 'Time-fractional diffusion: A discrete random walk approach', Nonlinear Dynamics 29, 2002, 129-143.
38. Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993 [translation from the Russian edition, Minsk, Nauka i Tekhnika, Minsk, 1987].
39. Uchaikin, V. V. and Zolotarev, V. M., Chance and Stability: Stable Distributions and their Applications, VSP, Utrecht, The Netherlands, 1999.
40. Feller, W., 'On a generalization of Marcel Riesz' potentials and the semi-groups generated by them', in Meddelanden Lunds Universitets Matematiska Seminarium, Lund, Sweden, 1952, pp. 73-81.
41. Gorenflo, R. and Mainardi, F., 'Random walk models for space-fractional diffusion processes', Fractional Calculus and Applied Analysis 1, 1998, 167-191.
42. Gorenflo, R. and Mainardi, F., 'Approximation of Lévy-Feller diffusion by random walk', Journal of Analysis and its Applications (ZAA) 18, 1999, 231-246.
43. Caputo, M., 'Linear models of dissipation whose Q is almost frequency independent, Part II', Geophysical Journal of the Royal Astronomical Society 13, 1967, 529-539.
44. Caputo, M., Elasticità e dissipazione, Zanichelli, Bologna, Italy, 1969 [in Italian].
45. Caputo, M. and Mainardi, F., 'Linear models of dissipation in anelastic solids', Rivista del Nuovo Cimento (Serie II) 1, 1971, 161-198.
46. Caputo, M., 'Models of flux in porous media with memory', Water Resources Research 36, 2000, 693-705.
47. Caputo, M., 'Distributed order differential equations modelling dielectric induction and diffusion', Fractional Calculus and Applied Analysis 4, 2001, 421-442.
48. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, California, 1999.
49. Diethelm, K. and Ford, N. J., 'Analysis of fractional differential equations', Journal of Mathematical Analysis and Applications 265, 2002, 229-248.
50. Diethelm, K., Ford, N. J., and Freed, A. D., 'A predictor-corrector method for the numerical solution of fractional differential equations', Nonlinear Dynamics 29, 2002, 3-22.
51. Djrbashian, M. M. and Nersesian A. B., 'Fractional derivatives and the Cauchy problem for differential equations of fractional order', Izvestia Academia Nauk Armjanskvy SSR, Matematika 3, 1968, 3-29 [in Russian].
52. Kochubei, A. N., 'A Cauchy problem for evolution equations of fractional order', Journal of Differential Equations 25, 1989, 967-974 [English translation from Russian].
53. Kochubei, A. N., 'Fractional order diffusion', Journal of Differential Equations 26, 1990, 485-492 [English translation from Russian].