On using random walks to solve the space-fractional advection-dispersion equations

Journal of Statistical Physics

Volumn 123, Issue 1, 2006, Pages 89-110

  Yong, Zhang ^a,b   Benson, David A ^b   Meerschaert, Mark M ^c   Scheffler, Hans Peter ^d  

^a DESERT RESEARCH INSTITUTE   (United States)

^b Department of Geology and Geological Engineering   (United States)

^c UNIVERSITY OF OTAGO   (New Zealand)

^d UNIVERSITY OF NEVADA   (United States)

Author keywords

Adjoint method; Forward equation; Fractional advection dispersion equation; Random walk

Indexed keywords

EID: 33646494026     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-006-9042-x     Document Type: Article

References (40)

1. B. Baeumer and M. M. Meerschaert, Frac. Calc. Appl. Anal 4:481 (2001).
2. D. A. Benson, Ph.D. dissertation, University of Nevada at Reno, 1998 (unpublished).
3. D. A. Benson, R. Schumer, M. M. Meerschaert, and S. W. Wheatcraft, Transp. Por. Media 42:211 (2001).
4. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, Water Resour. Res. 36(6):1403 (2000).
5. A. V. Chechkin, R. Gorenflo and I. M. Sokolov, J. Phys. A 38:L679 (2005).
6. A. V. Chechkin, V. Y. Gonchar, J. Klafter, R. Metzler, and L. V. Tanatarov, J. Sta. Phys. 115:1505 (2004).
7. S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, p. 380, John Wiley & Sons, New York, 1986.
8. W. Feller, An introduction to Probability Theory and Its Applications, second edition, John Wiley & Sons, New York, 1971.
9. R. Gorenflo and F. Mainardi, J. Anal. Appl. 18(2):231 (1999).
10. R. Gorenflo, A. Vivoli, and F. Mainardi, Nonlinear Dynamics 38:101 (2004).
11. R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi. Nonlinear Dynamics 29:129 (2002).
12. R. Gorenflo, G. D. Fabritiis, and F. Mainardi, Phys. A 269:79 (2004).
13. A. E. Hassan and M. M. Mohamed, J. Hyd. 275:242 (2002).
14. A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-stable Stochastic Processes, p. 355, Marcel Dekker, Inc., New York, 1994.
15. W. Kinzelbach, In: E. Custodio (Ed.), Groundwater Flow and Quality Modeling, pp. 227-245, Reidel Publishing Company, 1988.
16. K. A. Klise, V. C. Tidwell, S. A. McKenna, and M. D. Chapin, Geol. Soc. Am. Abstr. Programs 36(5):573 (2004).
17. E. M. LaBolle, G. E. Fogg, and A. F. B. Tompson, Water Resour. Res. 32:583 (1996).
18. E. M. LaBolle, J. Quastel, G. E. Fogg, and J. Gravner, Water Resour. Res. 36(3):651 (2000).
19. F. Liu, V. Anh, and I. Turner, J. Comput. Appl. Math. 166:209 (2004).
20. M. M. Meerschaert and C. Tadjeran, Appl. Nume. Math. 56:80 (2006).
21. M. M. Meerschaert and C. Tadjeran, J. of Comp. and Appl. Math. 172:65 (2004).
22. M. M. Meerschaert and H. P. Scheffler, Frac. Cal. Appl. Analy. 5(1):27 (2002).
23. M. M. Meerschaert and H. P. Scheffler, Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice, pp. 45-46, John Wiley & Sons, New York, 2001.
24. M. M. Meerschaert, D. A. Benson, and B. Baeumer, Phys. Rev. E 59(5):5026 (1999).
25. M. M. Meerschaert, D. A. Benson, and B. Baeumer, Phys. Rev. E 63(2):12 (2001).
26. M. M. Meerschaert, J. Mortensen, and S. W. Wheatcraft, Phys. A., to appear (2006).
27. R. Metzler and J. Klafter, J. Phys. A. 161:16 (2004).
28. R. Metzler, E. Barkai, and J. Klafter, Europhys. Lett. 46(4):431 (1999).
29. K. S. Miller and B. Ross. An Introduction to Fractional Calculus and Fractional Differential Equations, John Wiley, New York, 1993.
30. T. J. Osler, The Americ. Math. Mon. 78(6):645 (1971).
31. H. Risken, The Fokker-Planck Equation, p. 454, Springer & Verlag, New York, 1984.
32. J. P. Roop, Ph.D. dissertation, Clemson University, 2004 (unpublished).
33. G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall, New York, 1994.
34. H. Scher and M. Lax, Phys. Rev. B 7(10):4491 (1973).
35. R. Schumer, D. A. Benson, M. M. Meerschaert, and S. W. Wheatcraft, J. Contam. Hyd. 48:69 (2001).
36. I. M. Sokolov and R. Metzler, J. Phys. A 37: L609 (2004).
37. D. Stroock, Wahrscheinlichkeitstheorie verw. Gebiete 32:209 (1975).
38. G. J. M. Uffink, In: H. E. Kobus and W. Kinzelbach (Eds.), Contaminant Transport in Groundwater, p. 283, Brookfield: A.A. Balkema, Vt., 1989.
39. G. S. Weissmann, Y. Zhang, E. M. LaBolle, and G. E. Fogg, Water Resour. Res. 38(10), doi: 10.1029/2001WR000907.
40. V. V. Yanovsky, A. V. Chechkin, D. Schertzer, and A. V. Tur, Phys. A 282:13 (2000).