Some recent advances in theory and simulation of fractional diffusion processes

Journal of Computational and Applied Mathematics

Volumn 229, Issue 2, 2009, Pages 400-415

  Gorenflo, Rudolf ^a   Mainardi, Francesco ^b  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

^b UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

Anomalous diffusion; Fractional calculus; L vy processes; Parametric subordination; Power laws; Random walks; Renewal theory

Indexed keywords

ANOMALOUS DIFFUSION; FRACTIONAL CALCULUS; PARAMETRIC SUBORDINATION; POWER LAWS; RANDOM WALKS; RENEWAL THEORY;

DIFFUSION;

EID: 67349184476     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2008.04.005     Document Type: Article

References (48)

1. Andries E., Umarov S., and Steinberg S. Monte Carlo random walk simulation based on distributed order differential equations with applications to cell biology. Fractional Calculus and Applied Analysis 9 (2006) 351-370. http://arxiv.org/abs/math/0606797 [E-print http://arxiv.org/abs/math/0606797]
2. Baeumer B., and Meerschaert M.M. Stochastic solutions for fractional Cauchy problems. Fractional Calculus and Applied Analysis 4 (2001) 481-500
3. Barkai E. Fractional Fokker-Planck equation, solution, and application. Physical Review E 63 (2001) 046118-1/18
4. Bingham N.H., Goldie C.M., and Teugels J.L. Regular Variation (1987), Cambridge University Press, Cambridge
5. Chechkin A.V., Gorenflo R., and Sokolov I.M. Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Physical Review E 66 (2002) 046129/1-6
6. Chechkin A.V., Gorenflo R., Sokolov I.M., and Gonchar V.Yu. Distributed order time fractional diffusion equation. Fractional Calculus and Applied Analysis 6 (2002) 259-279
7. Cox D.R. Renewal Theory (1967), Methuen, London
8. Feller W. An Introduction to Probability Theory and its Applications vol. 2 (1971), Wiley, New York
9. Fulger D., Scalas E., and Germano G. Monte-Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Physical Review E 77 (2008). http://arxiv.org/abs/0707.2582 021122/1-7 [E-print http://arxiv.org/abs/0707.2582]
10. Gel̀fand I.M., and Shilov G.E. Generalized Functions vol. I (1964), Academic Press, New York and London
11. Gnedenko B.V., and Kolmogorov A.N. Limit Distributions for Sums of Independent Random Variables (1954), Addison-Wesley, Cambridge, Mass Translation from the Russian edition, State Publishing House for Technical-Theoretical Literature, Moscow and Leningrad, 1949
12. B.V. Gnedenko, I.N. Kovalenko, Introduction to Queueing Theory, Israel Program for Scientific Translations, Jerusalem (1968)
13. Gorenflo R., and Abdel-Rehim E. From power laws to fractional diffusion: The direct way. Vietnam Journal of Mathematics 32 SI (2004) 65-75
14. 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 Verlag, Wien 223-276. http://www.fracalmo.org [Reprinted in http://www.fracalmo.org]
15. Gorenflo R., and Mainardi F. Random walk models for space-fractional diffusion processes. Fractional Calculus and Applied Analysis 1 (1998) 167-191
16. R. Gorenflo, F. Mainardi, Non-Markovian random walks, scaling and diffusion limits, in: O.E. Barndorff-Nielsen (Eds.), Mini-Proceedings:e 2-nd MaPhySto Conference on Lévy Processes: Theory and Applications, Dept. Mathematics, University of Aarhus, Denmark, 21-25 January 2002, [Available c/o http://www.maphysto.dk, Miscellanea no. 22]
17. Gorenflo R., and Mainardi F. Simply and multiply scaled diffusion limits for continuous time random walks. In: Benkadda S., Leoncini X., and Zaslavsky G. (Eds). Proceedings of the International Workshop on Chaotic Transport and Complexity in Fluids and Plasmas Carry Le Rouet (France), 20-25 June 2004. Journal of Physics: Conference Series 7 (2005) 1-16
18. R. Gorenflo, F. Mainardi, Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects, in: R. Klages, G. Radons and I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim, Germany (2008) (in press) [E-print http://arxiv.org/abs/0705.0797]
19. Gorenflo R., Mainardi F., Scalas E., and Raberto M. Fractional calculus and continuous-time finance III: the diffusion limit. In: Kohlmann M., and Tang S. (Eds). Mathematical Finance (2001), Birkhäuser Verlag, Basel 171-180
20. Gorenflo R., Mainardi F., and Vivoli A. Continuous time random walk and parametric subordination in fractional diffusion. Chaos, Solitons and Fractals 34 (2007) 87-103. http://arxiv.org/abs/cond-mat/0701126 [E-print http://arxiv.org/abs/cond-mat/0701126]
21. Hilfer R., and Anton L. Fractional master equations and fractal time random walks. Physical Review E 51 (1995) R848-R851
22. Ilic M., Liu F., Turner I., and Anh V. Numerical approximation of a fractional-in-space diffusion equation (II) with non-homogeneous boundary conditions. Fractional Calculus and Applied Analysis 9 (2006) 333-349
23. Jacob N. Pseudodifferential Operators - Markov Processes. Vol I: Fourier Analysis and Semigroups Vol II: Generators and Their Potential Theory, Vol. III: Markov Processes and Applications (2001), Imperial College Press, London (2002), (2005)
24. A. Janicki, Numerical and Statistical Approximation of Stochastic Differential Equations with Non-Gaussian Measures, Monograph No 1, H. Steinhaus Center for Stochastic Methods in Science and Technology, Technical University, Wroclaw, Poland 1996
25. Janicki A., and Weron A. Simulation and Chaotic Behavior of α-Stable Stochastic Processes (1994), Marcel Dekker, New York
26. Kilbas A.A., Srivastava H.M., and Trujillo J.J. Theory and Applications of Fractional Differential Equations (2006), Elsevier, Amsterdam
27. Kleinhans D., and Friedrich R. Continuous time random walks: simulation of continuous trajectories. Physical Review E 76 (2008). http://arxiv/abs/0707.3221 061102/1-6 [E-print http://arxiv/abs/0707.3221]
28. 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. http://arxiv.org/abs/cond-mat/0702419 [E-print http://arxiv.org/abs/cond-mat/0702419]
29. Mainardi F., Pagnini G., and Saxena R.K. Fox H functions in fractional diffusion. Journal of Computational and Applied Mathematics 178 (2005) 321-331
30. 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
31. 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
32. 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
33. Meerschaert M.M., and Scheffler H.-P. Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice (2001), Wiley Interscience, New York
34. Metzler R., and Klafter J. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Physical Reports 339 (2000) 1-77
35. Montroll E.W., and Scher H. Random walks on lattices, IV: Continuous-time walks and influence of absorbing boundaries. Journal of Statistical Physics 9 (1973) 101-135
36. Montroll E.W., and Weiss G.H. Random walks on lattices, II. Journal of Mathematical Physics 6 (1965) 167-181
37. Newman M.E.J. Power laws, Pareto distributions and Zipf's law. Contemporary Physics 46 (2005) 323-351. http://arxiv.org/abs/cond-mat/0412004 [E-print http://arxiv.org/abs/cond-mat/0412004]
38. Podlubny I. Fractional Differential Equations (1999), Academic Press, San Diego
39. Piryatinska A., Saichev A.I., and Woyczynski W.A. Models of anomalous diffusion: The subdiffusive case. Physica A 349 (2005) 375-420
40. Samko S.G., Kilbas A.A., and Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach, New York
41. Sato K.-I. Lévy Processes and Infinitely Divisible Distributions (1999), Cambridge University Press, Cambridge
42. Scalas E. The application of continuous-time random walks in finance and economics. Physica A 362 (2006) 225-239
43. Scalas E., Gorenflo R., and Mainardi F. Fractional calculus and continuous-time finance. Physica A 284 (2000) 376-384
44. 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
45. Schroeder M. Fractals, Chaos, Power Laws (1991), Freeman, New York
46. Sokolov I.M., Chechkin A.V., and Klafter J. Distributed-order fractional kinetics. Acta Physica Polonica 35 (2004) 1323-1341
47. Sokolov I.M., and Klafter J. From diffusion to anomalous diffusion: a century after Einstein's Brownian motion. Chaos 15 (2005) 026103-026109
48. Weiss G.H. Aspects and Applications of Random Walks (1994), North-Holland, Amsterdam