Time Fractional Diffusion: A Discrete Random Walk Approach

Springer Science and Business Media LLC - Tập 29 - Trang 129-143 - 2002
Rudolf Gorenflo1, Francesco Mainardi2, Daniele Moretti3, Paolo Paradisi4
1Erstes Mathematisches Institut, Freie Universität Berlin, Berlin, Germany
2Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, Bologna, Italy
3CRIBISNET S.p.A., Bologna, Italy
4ISAC, Istituto di Scienze dell'Atmosfera e del Clima, Sezione di Lecce, Lecce, Italy

Tóm tắt

The time fractional diffusion equation is obtained from the standarddiffusion equation by replacing the first-order time derivative with afractional derivative of order β ∋ (0, 1). From a physicalview-point this generalized diffusion equation is obtained from afractional Fick law which describes transport processes with longmemory. The fundamental solution for the Cauchy problem is interpretedas a probability density of a self-similar non-Markovian stochasticprocess related to a phenomenon of slow anomalous diffusion. By adoptinga suitable finite-difference scheme of solution, we generate discretemodels of random walk suitable for simulating random variables whosespatial probability density evolves in time according to this fractionaldiffusion equation.

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Tài liệu tham khảo

Mainardi, F., 'On the initial value problem for the fractional diffusion-wave equation', in Waves and Stability in Continuous Media, S. Rionero and T. Ruggeri (eds.), World Scientific, Singapore, 1994, pp. 246–251.

Mainardi, F., 'Fractional relaxation-oscillation and fractional diffusion-wave phenomena', Chaos, Solitons and Fractals 7, 1996, 1461–1477.

Mainardi, F., 'Fractional calculus: Some basic problems in continuum and statistical mechanics', in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (eds.), Springer Verlag, Wien, 1997, pp. 291–248. [Reprinted in NEWS 011201, see http://www.fracalmo.org]

Metzler, R., Glöckle, W.G. and Nonnenmacher, T.F., 'Fractional model equation for anomalous diffusion', Physica A 211, 1994, 13–24.

Feller, W., 'On a generalization of Marcel Riesz' potentials and the semi-groups generated by them', in Meddelanden Lunds Universitets Matematiska Seminarium, Lund, 1952, pp. 73–81.

West, B.J. and Seshadri, V., 'Linear systems with Lévy fluctuations', Physica A 113, 1982, 203–216.

Gorenflo, R. and Mainardi, F., 'Random walk models for space fractional diffusion processes', Fractional Calculus and Applied Analysis 1, 1998, 167–191.

Gorenflo, R. and Mainardi, F., 'Approximation of Lévy-Feller diffusion by random walk', Journal for Analysis and its Applications (ZAA) 18, 1999, 231–146.

Gorenflo, R. and Mainardi, F., Random walk models approximating symmetric space fractional diffusion processes, in Problems in Mathematical Physics, J. Elschner, I. Gohberg and B. Silbermann (eds.), Operator Theory: Advances and Applications, Vol. 121, Birkhäuser, Basel, 2001, pp. 120–145.

Gorenflo, R., De Fabritiis, G., and Mainardi, F., 'Discrete random walk models for symmetric Lévy-Feller diffusion processes', Physica A 269, 1999, 79–89.

Paradisi, P., Cesari, R., Mainardi, F., and Tampieri, F., 'The fractional Fick's law for non-local transport processes', Physica A 293, 2001, 130–142.

Saichev, A. and Zaslavsky, G., 'Fractional kinetic equations: Solutions and applications', Chaos 7, 1997, 753–764.

West, B. J., Grigolini, P., Metzler, R., and Nonnenmacher, T. F., 'Fractional diffusion and Lévy stable processes', Physical Review E 55, 1997, 99–106.

Anh, V. V. and Leonenko, N. N., 'Spectral analysis of fractional kinetic equations with random data', Journal of Statistical Physics 104, 2001, 1349–1387.

Gorenflo, R., Iskenderov, A., and Luchko, Yu., 'Mapping between solutions of fractional diffusion-wave equations', Fractional Calculus and Applied Analysis 3, 2000, 75–86.

Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., and Paradisi, P., 'Fractional diffusion: Probability distributions and random walk models', Physica A 305, 2002, 106–112.

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]

Montroll, E. W. and Weiss, G. H., 'Random walks on lattices, II', Journal of Mathematical Physics 6, 1965, 167–181.

Montroll, E. W. and West, B. J., 'On an enriched collection of stochastic processes', in Fluctuation Phenomena, E. W. Montroll and J. Leibowitz (eds.), North-Holland, Amsterdam, 1979, pp. 61–175.

Montroll, E. W. and Shlesinger, M. F., 'On the wonderful world of random walks', in Nonequilibrium Phenomena II: From Stochastics to Hydrodynamics, J. Leibowitz and E.W. Montroll (eds.), North-Holland, Amsterdam, 1984, pp. 1–121.

Hilfer, R. and Anton, L., 'Fractional master equations and fractal time random walks', Physical Review E 51, 1995, R848–R851.

Compte, A., 'Stochastic foundations of fractional dynamics', Physical Review E 53, 1996, 4191–4193.

Uchaikin, V. V. and Zolotarev, V. M., Chance and Stability. Stable Distributions and Their Applications, VSP, Utrecht, 1999.

Uchaikin, V.V., 'Montroll-Weiss problem, fractional equations and stable distributions', International Journal of Theoretical Physics 39, 2000, 2087–2105.

Hilfer, R., 'Fractional time evolution', in Applications of Fractional Calculus in Physics, R. Hilfer (ed.), World Scientific, Singapore, 2000, pp. 87–130.

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.

Scalas, E., Gorenflo, R., and Mainardi, F., 'Fractional calculus and continuous-time finance', Physica A 284, 2000, 376–384.

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.

Gorenflo, R., Mainardi, F., Scalas, E., and Raberto, M., 'Fractional calculus and continuous-time finance III: The diffusion limit', in Mathematical Finance, M. Kohlmann and S. Tang (eds.), Birkhäuser Verlag, Basel, 2001, pp. 171–180.

Metzler, R. and Klafter, J., 'The random walk's guide to anomalous diffusion: a fractional dynamics approach', Physics Reports 339, 2000, 1–77.

Klafter, J., Shlesinger, M.F., and Zumofen, G., 'Beyond Brownian motion', Physics Today 49, 1996, 33–39.

Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, CA, 1999.

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 (1987)]

Gorenflo, R., 'Nichtnegativitäts-und substanzerhaltende Differenzenschemata für lineare Diffusionsgleichungen', Numerische Mathematik 14, 1970, 448–467.

Gorenflo, R., 'Conservative difference schemes for diffusion problems', in Behandlung von Differentialgleichungen mit besonderer Berücksichtigung freier Randwertaufgaben, J. Albrecht, L. Collatz, and G. Hämmerlin (eds.), International Series of Numerical Mathematics, Vol. 39 Birkhäuser, Basel, 1978, pp. 101–124.

Gorenflo, R. and Mainardi, F., 'Fractional calculus: Integral and differential equations of fractional order', in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (eds.), Springer-Verlag, Wien, 1997, pp. 223–276. [Reprinted in NEWS 010101, see http://www.fracalmo.org]

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.

Caputo, M., Elasticità e Dissipazione, Zanichelli, Bologna, 1969 [in Italian].

Caputo, M. and Mainardi, F., 'Linear models of dissipation in anelastic solids', Rivista del Nuovo Cimento (Ser. II) 1, 1971, 161–198.

Feller,W., An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, New York, 1971.

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Springer Science and Business Media LLC

Tập 29

129-143

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