Stochastic solution of space-time fractional diffusion equations

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 65, Issue 4, 2002, Pages 4-

  Meerschaert, Mark M ^a   Benson, David A ^b   Scheffler, Hans Peter ^c   Baeumer, Boris ^d  

^a UNIVERSITY OF NEVADA   (United States)

^b DESERT RESEARCH INSTITUTE   (United States)

^c TU DORTMUND UNIVERSITY   (Germany)

^d UNIVERSITY OF OTAGO   (New Zealand)

Author keywords

[No Author keywords available]

Indexed keywords

MULTISCALING SPACE FRACTIONAL DERIVATIVES; PSEUDODIFFERENTIAL OPERATORS; SPACE TIME FRACTIONAL DIFFUSION EQUATIONS; STOCHASTIC SOLUTION;

BROWNIAN MOVEMENT; HAMILTONIANS; LAPLACE TRANSFORMS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; PROBABILITY DISTRIBUTIONS; RANDOM PROCESSES;

DIFFUSION;

ARTICLE;

EID: 41349099804     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.65.041103     Document Type: Article

References (43)

1. S. Bochner, Proc. Natl. Acad. Sci. U.S.A. 35, 368 (1949).
2. A. Compte, Phys. Rev. E 53, 4191 (1996).
3. A. Compte, Phys. Rev. E 55, 6821 (1997).
4. E. Barkai, Phys. Rev. E 63, 046118 (2001).
5. R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000).
6. An excellent review is given by R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
7. R. Lenormand and B. Wang, Transp. Porous Media 18, 263 (1995);
8. D. Benson, S. Wheatcraft, and M. Meerschaert, Water Resour. Res. 36, 1403 (2000);
9. D. Benson, et al., Transp. Porous Media 42, 211 (2001);
10. B. Baeumer, et al., Water Resour. Res. 37, 1543 (2001).
11. Y. Hatano and N. Hatano, Water Resour. Res. 34, 1027 (1998)
12. R. Haggerty, S. McKenna, and L. Meigs, Water Resour. Res. 36, 3467 (2000).
13. M. Meerschaert and H.P. Scheffler, Limit Theorems for Sums of Independent Random Vectors (Wiley, New York, 2001).
14. M.M. Meerschaert, D.A. Benson, and B. Baumer, Phys. Rev. E 59, 5026 (1999).
15. M.M. Meerschaert, D.A. Benson, and B. Baeumer, Phys. Rev. E 63, 021112 (2001).
16. J. Bertoin, Lévy Processes (Cambridge University Press, Cambridge, 1996).
17. K.I. Sato, Lévy Processes and Infinitely Divisible Distributions (Cambridge University Press, Cambridge, 1999).
18. W. Feller, An Introduction to Probability Theory and Its Applications, 2nd ed. (Wiley, New York, 1971), Vol. II.
19. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences Vol. 44 (Springer, New York, 1983).
20. W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics (Birkhaeuser-Verlag, Berlin, 2001).
21. N. Bingham, Z. Wahrscheinlichkeitstheor. Verwandte. Geb. 17, 1 (1971).
22. L. Bondesson, G. Kristiansen, and F. Steutel, Statist. Probab. Lett. 28, 271 (1996).
23. E. Barkai, R. Metzler, and J. Klafter, Phys. Rev. E 61, 132 (2000).
24. M. Kotulski and K. Weron, Lecture Notes in Statistics (Springer, New York, 1995), Vol. 114, p. 379.
25. E.W. Montroll and G.H. Weiss, J. Math. Phys. 6, 167 (1965).
26. H. Scher and M. Lax, Phys. Rev. B 7, 4491 (1973).
27. J. Klafter, A. Blumen, and M.F. Shlesinger, Phys. Rev. A 35, 3081 (1987).
28. V.V. Uchaikin and V.M. Zolotarev, Chance and Stability. Stable Distributions and Their Applications (VSP, Utrecht, 1999).
29. M. Kotulski, J. Stat. Phys. 81, 777 (1995).
30. E. Barkai, e-print cond-mat/0108024.
31. M. Shlesinger, J. Klafter, and Y.M. Wong, J. Stat. Phys. 27, 499 (1982).
32. M. Meerschaert and H. P. Scheffler (unpublished).
33. B. Mandelbrot and J.W. Van Ness, SIAM Rev. 10, 422 (1968).
34. D. Sornette, Phys. Rep. 297, 5 (1998).
35. W. Wyss, J. Stat. Phys. 27, 2782 (1986).
36. W.R. Schneider and W. Wyss, J. Math. Phys. 30, 134 (1989).
37. G. Zaslavsky, Physica D 76, 110 (1994).
38. A. Saichev and G. Zaslavsky, Chaos 7, 753 (1997).
39. A. Chaves, Phys. Lett. A 239, 13 (1998).
40. D. Benson, S. Wheatcraft, and M. Meerschaert, Water Resour. Res. 36, 1413 (2000).
41. R. Metzler, J. Klafter, and I.M. Sokolov, Phys. Rev. E 58, 1621 (1998).
42. Z. Jurek and J.D. Mason, Operator-Limit Distributions in Probability Theory (Wiley, New York, 1993).
43. B. Baeumer and M. Meerschaert, Fractional Calculus and Appl. Analysis 4, 481(2001).