Generalized fractional diffusion equations for accelerating subdiffusion and truncated Lévy flights

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 78, Issue 2, 2008, Pages

  Chechkin, A V ^a   Gonchar, V Yu ^a   Gorenflo, R ^b   Korabel, N ^c   Sokolov, I M ^d  

^a KHARKOV INSTITUTE OF PHYSICS AND TECHNOLOGY   (Ukraine)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

^c BAR ILAN UNIVERSITY   (Israel)

^d HUMBOLDT UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC ANALYSIS; DIFFUSION; LINEAR EQUATIONS; MARKOV PROCESSES;

ANOMALOUS DIFFUSIONS; ASYMPTOTICS; DIFFUSION EQUATIONS; DIFFUSION EXPONENT; DIFFUSION PHENOMENA; FRACTIONAL DERIVATIVES; FRACTIONAL DIFFUSION EQUATIONS; GAUSSIAN SHAPES; GENERAL SITUATION; POWER LAWS; STABLE DISTRIBUTIONS; SUBDIFFUSION; SUPERDIFFUSION;

SEMICONDUCTOR DOPING;

EID: 49549105572     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.78.021111     Document Type: Article

References (62)

1. R. Metzler and J. Klafter, Phys. Rep. PRPLCM 0370-1573 10.1016/S0370-1573(00)00070-3 339, 1 (2000).
2. I. M. Sokolov, J. Klafter, and A. Blumen, Phys. Today PHTOAD 0031-9228 55 (11), 48 (2002).
3. I. M. Sokolov, A. V. Chechkin, and J. Klafter, Acta Phys. Pol. B APOBBB 0587-4254 35, 1323 (2004).
4. R. N. Mantegna and H. E. Stanley, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.73.2946 73, 2946 (1994).
5. I. Koponen, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.52.1197 52, 1197 (1995).
6. Ya. G. Sinai, Theory Probab. Appl. 27, 256 (1983). 0040-585X
7. I. M. Sokolov, A. V. Chechkin, and J. Klafter, Physica A PHYADX 0378-4371 10.1016/j.physa.2003.12.044 336, 245 (2004).
8. A. V. Chechkin, J. Klafter, and I. M. Sokolov, Europhys. Lett. EULEEJ 0295-5075 10.1209/epl/i2003-00539-0 63, 326 (2003).
9. M. Caputo, Elasticità e Dissipazione (Zanichelli Printer, Bologna, 1969).
10. M. Caputo, Ann. Univ. Sez. VII, Sc. Mat. ZZZZZZ 1069-5869 XLI, 73 (1995).
11. M. Caputo, Fractional Calculus Appl. Anal. ZZZZZZ 1311-0454 4, 421 (2001).
12. A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.66.046129 66, 046129 (2002).
13. I. M. Sokolov and J. Klafter, Chaos CHAOEH 1054-1500 10.1063/1.1860472 15, 026103 (2005).
14. A. V. Chechkin, R. Gorenflo, I. M. Sokolov, and V. Yu. Gonchar, Fractional Calculus Appl. Anal. ZZZZZZ 1311-0454 6, 259 (2003).
15. S. Umarov and R. Gorenflo, Z. Anal. Anw. 0165-2370 24, 449 (2005).
16. S. Umarov and S. Steinberg, IMS Lecture Notes-Monograph Series Vol. 51: High Dimensional Probablility (Institute of Mathematical Physics, Bethesda, MD, 2006), pp. 117 - 127.
17. A. N. Kochubei, e-print arXiv:math-ph/0703046v1, J. Math. Anal. Appl. (to be published)
18. e-print arXiv:0710.1710v1, J. Ukr. Mth. J. (to be published).
19. I. M. Sokolov, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.63.056111 63, 056111 (2001).
20. P. Paradisi, R. Cesari, F. Mainardi, and F. Tampieri, Physica A PHYADX 0378-4371 10.1016/S0378-4371(00)00491-X 293, 130 (2001).
21. R. Gorenflo and F. Mainardi, in Fractals and Fractional Calculus in Continuum Mechanics CISM Courses and Lectures No. 378, edited by, A. Carpinteri, and, F. Mainardi, (Springer-Verlag, Wien, New York, 1997), pp. 223-276.
22. F. Mainardi, A. Mura, G. Pagnini, and R. Gorenflo, e-print arXiv:cond-mat/0701132v1, J. Vib. Control (to be published).
23. F. Mainardi, A. Mura, G. Pagnini, and R. Gorenflo, in Mathematical Methods in Engineering, edited by, K. Tas, J. A. Tenreiro-Machado, and, D. Baleanu, (Springer-Verlag, Dordrecht, 2007), pp. 23-55.
24. T. A. M. Langlands, Physica A PHYADX 0378-4371 10.1016/j.physa.2005.12. 012 367, 136 (2006).
25. W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1971), Vol. 2.
26. P. P. Valko and J. Abate, Comput. Math. Appl. CMAPDK 0898-1221 10.1016/j.camwa.2002.10.017 48, 629 (2004)
27. J. Abate and P. P. Valko Int. J. Numer. Methods Eng. IJNMBH 0029-5981 10.1002/nme.995 60, 979 (2004)
28. P. Valko and J. Abate, Appl. Numer. Math. ANMAEL 0168-9274 10.1016/j.apnum.2004.10.002 53, 73 (2005).
29. F. Mainardi and M. Tomirotti, in Transform Methods and Special Functions, edited by, P. Rusev, I. Dimovski, and, V. Kiryakova, Science Culture Technology, Singapore, 1995), pp. 171-183.
30. I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, CA, 1998).
31. G. Zumofen and J. Klafter, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.47.851 47, 851 (1993).
32. N. Mordant, P. Metz, O. Michel, and J.-F. Pinton, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.87.214501 87, 214501 (2001)
33. N. Mordant, J. Delour, E. Léveque, A. Arnéodo, and J.-F. Pinton, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.89.254502 89, 254502 (2002)
34. N. Mordant, J. Delour, E. Léveque, O. Michel, A. Arnéodo, and J.-F. Pinton, J. Stat. Phys. JSTPBS 0022-4715 10.1023/A:1027352417365 113, 701 (2003).
35. R. Friedrich, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.90. 084501 90, 084501 (2003).
36. S. Khan, A. M. Reynolds, I. E. G. Morrison, and R. J. Cherry, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.71.041915 71, 041915 (2005).
37. A. M. Reynolds, Phys. Lett. A PYLAAG 0375-9601 10.1016/j.physleta.2005. 05.086 342, 439 (2005).
38. M. J. Saxton, Biophys. J. BIOJAU 0006-3495 10.1529/biophysj.106.092619 92, 1178 (2007).
39. A. L. Hodgkin and R. D. Keynes, J. Physiol. 128, 61 (1955). 0022-3751
40. J. Kärger and D. M. Ruthven, Diffusion in Zeolites and Other Microporous Solids (Wiley, New York, 1992).
41. P. Demontis, G. Stara, and G. B. Suffritti, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.1697382 120, 9233 (2004).
42. P. Demontis and G. B. Suffritti, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.74.051112 74, 051112 (2006).
43. T. Kosztolowicz, K. Dworecki, and St. Mrowczynski, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.94.170602 94, 170602 (2005).
44. T. Kosztolowicz, K. Dworecki, and St. Mrowczynski, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.71.041105 71, 041105 (2005).
45. A. I. Saichev and G. M. Zaslavsky, Chaos CHAOEH 1054-1500 10.1063/1.166272 7, 753 (1997).
46. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications (Nauka i Technica, Minsk, 1987), in Russian (English translation: Gordon and Breach, Amsterdam, 1993).
47. R. Gorenflo and F. Mainardi, in Problems in Mathematical Physics, Operator Theory: Advances and Applications No. 121 (Siegfried Prössdorf Memorial Volume), edited by, J. Elschner, I. Gohberg, and, B. Silbermann, (Birkhäuser-Verlag, Basel, Switzerland, 2001), pp. 120-145.
48. G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes (Chapman & Hall, New York, 1994).
49. G. H. Hardy, Divergent Series (Clarendon, Oxford, 1949).
50. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Chelsea, New York, 1986).
51. B. J. West and V. Seshadri, Physica A PHYADX 0378-4371 10.1016/0378-4371(82)90015-2 113, 203 (1982).
52. Gy. Terdik, W.A. Woyczynski, and A. Piryatinska, Phys. Lett. A PYLAAG 0375-9601 10.1016/j.physleta.2005.08.083 348, 94 (2006).
53. R. Jha, P.K. Kaw, D.R. Kulkarni, J.C. Parikh, and ADITYA Team, Phys. Plasmas PHPAEN 1070-664X 10.1063/1.1541607 10, 699 (2003).
54. V. Yu. Gonchar, A. V. Chechkin, E. D. Sorokovoi, V. V. Chechkin, L. I. Grigor'eva, and E. D. Volkov, Plasma Phys. Rep. PPHREM 1063-780X 10.1134/1.1575308 29, 380 (2003).
55. T. Mizuuchi, J. Nucl. Mater. 337-339, 332 (2005). 0022-3115
56. V. Budaev, Y. Kikuchi, Y. Uesugi, and S. Takamura, Nucl. Fusion NUFUAU 0029-5515 10.1088/0029-5515/44/6/S11 44, S108 (2004).
57. H. E. Stanley, Physica A PHYADX 0378-4371 10.1016/S0378-4371(02)01825-3 318, 279 (2003).
58. R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge University Press, Cambridge, England, 2000).
59. R. N. Mantegna and H. E. Stanley, Nature (London) NATUAS 0028-0836 10.1038/376046a0 376, 46 (1995).
60. P. Gopikrishnan, M. Meyer, L. A. N. Amaral, and H. E. Stanley, Eur. Phys. J. B EPJBFY 1434-6028 10.1007/s100510050292 3, 139 (1998).
61. X. Gabaix, P. Gopikrishnan, V. Plerou, and H. E. Stanley, Nature (London) NATUAS 0028-0836 10.1038/nature01624 423, 267 (2003).
62. M. H. Cohen and P. Venkatesh, in Practical Fruits of Econophysics. Proceedings of the Third Nikkei Econophysics Symposium, edited by, H. Takayasu, (Springer-Verlag, Tokyo, 2006), pp. 147-151).