The M -Wright function in time-fractional diffusion processes: A tutorial survey

International Journal of Differential Equations

Volumn 2010, Issue , 2010, Pages

  Mainardi, Francesco ^a   Mura, Antonio ^b   Pagnini, Gianni ^c  

^a UNIVERSITY OF BOLOGNA   (Italy)

^b Angelini Ricerche SpA   (Italy)

^c CRS4   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 78650789902     PISSN: 16879643     EISSN: 16879651     Source Type: Journal    
DOI: 10.1155/2010/104505     Document Type: Review

References (79)

1. Mandelbrot B. B., Van Ness J. W., Fractional Brownian motions, fractional noises and applications. SIAM Review 1968 10 422 437 MR0242239 10.1137/1010093 ZBL0179.47801
2. Taqqu M. S., Doukan P., Oppenheim G., Taqqu M. S., Fractional Brownian motion and long-range dependence. Theory and Applications of Long-Range Dependence 2003 Boston, Mass, USA Birkhäuser 5 38 MR1956042 ZBL1039.60041
3. Gorenflo R., Mainardi F., Vivoli A., Continuous-time random walk and parametric subordination in fractional diffusion. Chaos, Solitons & Fractals 2007 34 1 87 103 10.1016/j.chaos.2007.01.052 MR2320430 ZBL1142.82363 (Pubitemid 46529249)
4. Meerschaert M. M., Benson D. A., Scheffler H.-P., Baeumer B., Stochastic solution of space-time fractional diffusion equations. Physical Review E 2002 65 4 4 041103 10.1103/PhysRevE.65.041103 MR1917983 (Pubitemid 40618799)
5. Metzler R., Klafter J., The random walk's guide to anomalous diffusion: a fractional dynamics approach. Physics Reports A 2000 339 1 1 77 10.1016/S0370-1573(00)00070-3 MR1809268 ZBL0984.82032
6. Scalas E., Gorenflo R., Mainardi F., Uncoupled continuous-time random walks: solution and limiting behavior of the master equation. Physical Review E 2004 69 1 8 011107 10.1103/PhysRevE.69.011107 MR2097999
7. Dubkov A. A., Spagnolo B., Uchaikin V. V., Lévy flight superdiffusion: an introduction. International Journal of Bifurcation and Chaos 2008 18 9 2649 2672 10.1142/S0218127408021877 MR2479323 ZBL1157.82300
8. Schneider W. R., Albeverio S., Casati G., Cattaneo U., Merlini D., Moresi R., Grey noise. Stochastic Processes, Physics and Geometry (Ascona and Locarno, 1988) 1990 Teaneck, NJ, USA World Scientific 676 681 MR1124240
9. Schneider W. R., Albeverio S., Fenstad J. E., Holden H., Lindstrøm T., Grey noise. Ideas and Methods in Mathematical Analysis, Stochastics, and Applications (Oslo, 1988) 1992 Cambridge, UK Cambridge University Press 261 282 MR1190506 ZBL0761.60036
10. Chechkin A. V., achechkin@kipt.kharkov.ua Gorenflo R., gorenflo@math.fu-berlin.de Sokolov I. M., igor.sokolov@physik.hu-berlin.de Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Physical Review E 2002 66 4 7 046129 2-s2.0-6744272549 10.1103/PhysRevE.66.046129 (Pubitemid 40606254)
11. Chechkin A. V., Gorenflo R., Sokolov I. M., Gonchar V. Yu., Distributed order time fractional diffusion equation. Fractional Calculus & Applied Analysis 2003 6 3 259 279 MR2035651 ZBL1089.60046
12. Chechkin A. V., Gonchar V. Yu., Gorenflo R., Korabel N., Sokolov I. M., Generalized fractional diffusion equations for accelerating subdiffusion and truncated Lévy flights. Physical Review E 2008 78 2 13 021111 10.1103/PhysRevE.78.021111 MR2496818
13. Gorenflo R., Mainardi F., Moretti D., Paradisi P., Time fractional diffusion: a discrete random walk approach. Nonlinear Dynamics 2002 29 1-4 129 143 10.1023/A:1016547232119 MR1926470 ZBL1009.82016 (Pubitemid 34945395)
14. Gorenflo R., Mainardi F., mainardi@bo.infn.it Moretti D., Pagnini G., Paradisi P., Discrete random walk models for space-time fractional diffusion. Chemical Physics 2002 284 1-2 521 541 2-s2.0-0032049827 10.1016/S0301-0104(02) 00714-0
15. Liu F., Shen S., Anh V., Turner I., Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. The ANZIAM Journal 2005 46 C488 C504 MR2182183
16. Liu F., Zhuang P., Anh V., Turner I., Burrage K., Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Applied Mathematics and Computation 2007 191 1 12 20 10.1016/j.amc.2006.08.162 MR2385498 (Pubitemid 47223382)
17. Zhang Y., Benson D. A., Meerschaert M. M., Scheffler H.-P., On using random walks to solve the space-fractional advection-dispersion equations. Journal of Statistical Physics 2006 123 1 89 110 10.1007/s10955-006-9042-x MR2225237 ZBL1092.82038 (Pubitemid 43699923)
18. Zhuang P., Liu F., Implicit difference approximation for the time fractional diffusion equation. Journal of Applied Mathematics & Computing 2006 22 3 87 99 10.1007/BF02832039 MR2323899 ZBL1140.65094 (Pubitemid 44695544)
19. Mura A., Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis, Ph.D. thesis March 2008 Department of Physics, University of Bologna. Supervisor Prof. F. Mainardi, http://www.fracalmo.org/mura/
20. Mura A., Mainardi F., A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics. Integral Transforms and Special Functions 2009 20 3-4 185 198 10.1080/10652460802567517 MR2501791 ZBL1173.26005
21. Mura A., Pagnini G., Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. Journal of Physics A 2008 41 28 22 285003 10.1088/1751-8113/41/28/285003 MR2430462 ZBL1143.82028
22. Mura A., mura@bo.infn.it Taqqu M. S., Mainardi F., Non-Markovian diffusion equations and processes: analysis and simulations. Physica A 2008 387 21 5033 5064 2-s2.0-36149013316 10.1016/j.physa.2008.04.035
23. Caputo M., Linear models of dissipation whose Q is almost frequency independent-part II. Geophysical Journal of the Royal Astronomical Society 1967 13 529 539
24. Caputo M., Elasticity and Dissipation 1969 Bologna, Italy Zanichelli
25. Gorenflo R., Mainardi F., Carpinteri A., Mainardi F., Fractional calculus: integral and differential equations of fractional order. Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996) 1997 378 Vienna, Austria Springer 223 276 CISM Courses and Lectures MR1611585
26. Mainardi F., Gorenflo R., Time-fractional derivatives in relaxation processes: a tutorial survey. Fractional Calculus & Applied Analysis 2007 10 3 269 308 MR2382782 ZBL1157.26304
27. Podlubny I., Fractional Differential Equations 1999 198 San Diego, Calif, USA Academic Press xxiv+340 Mathematics in Science and Engineering MR1658022 ZBL0924.34008
28. Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations 2006 204 Amsterdam, The Netherlands Elsevier Science B.V. xvi+523 North-Holland Mathematics Studies MR2218073 ZBL1092.45003
29. Wright E. M., On the coefficients of power series having exponential singularities. Journal of the London Mathematical Society 1933 8 71 79
30. Wright E. M., The asymptotic expansion of the generalized Bessel function. Proceedings of the London Mathematical Society 1935 38 257 270
31. Wright E. M., The asymptotic expansion of the generalized hypergeometric function. Journal of the London Mathematical Society 1935 10 287 293
32. Wright E. M., The generalized Bessel function of order greater than one. The Quarterly Journal of Mathematics 1940 11 36 48 MR0003875 ZBL0023.14101
33. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher Transcendental Functions 1954 3 chapter 18 New-York, NY, USA McGraw-Hill
34. Gorenflo R., Luchko Yu., Mainardi F., Analytical properties and applications of the Wright function. Fractional Calculus & Applied Analysis 1999 2 4 383 414 MR1752379 ZBL1027.33006
35. Gorenflo R., Luchko Yu., Mainardi F., Wright functions as scale-invariant solutions of the diffusion-wave equation. Journal of Computational and Applied Mathematics 2000 118 1-2 175 191 10.1016/S0377-0427(00)00288-0 MR1765948 ZBL0973.35012
36. Kilbas A. A., Saigo M., Trujillo J. J., On the generalized Wright function. Fractional Calculus & Applied Analysis 2002 5 4 437 460 MR1967844 ZBL1027.33015
37. Kiryakova V., Generalized Fractional Calculus and Applications 1994 301 Harlow, UK Longman Scientific & Technical x+388 Pitman Research Notes in Mathematics Series MR1265940 ZBL0882.26003
38. Mainardi F., Carpinteri A., Mainardi F., Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996) 1997 378 Vienna, Austria Springer 291 348 CISM Courses and Lectures MR1611587 ZBL0917.73004
39. Stanković B., On the function of E. M. Wright. Publications de l'Institut Mathèmatique 1970 10 (24) 113 124 MR0280762 ZBL0204.08404
40. Wong R., Zhao Y.-Q., Smoothing of Stokes's discontinuity for the generalized Bessel function. Proceedings of the Royal Society A 1999 455 1984 1381 1400 10.1098/rspa.1999.0365 MR1701756 ZBL0974.41021
41. Wong R., Zhao Y.-Q., Smoothing of Stokes's discontinuity for the generalized Bessel function. II. Proceedings of the Royal Society A 1999 455 1988 3065 3084 10.1098/rspa.1999.0440 MR1807056 ZBL0974.41021
42. Mainardi F., Rionero S., Ruggeri T., On the initial value problem for the fractional diffusion-wave equation. Waves and Stability in Continuous Media (Bologna, 1993) 1994 23 River Edge, NJ, USA World Scientific 246 251 Series on Advances in Mathematics for Applied Sciences MR1320083
43. Mainardi F., Tomirotti M., Rusev P., Dimovski I., Kiryakova V., On a special function arising in the time fractional diffusion-wave equation. Transform Methods and Special Functions, Sofia 1994 1995 Singapore Science Culture Technology 171 183
44. Bender C. M., Orszag S. A., Advanced Mathematical Methods for Scientists and Engineers 1987 Singapore McGraw-Hill
45. Hanyga A., Multi-dimensional solutions of space-time-fractional diffusion equations. Proceedings of the Royal Society A 2002 458 2018 429 450 10.1098/rspa.2001.0893 MR1889936 ZBL0999.60035
46. Balescu R., V -Langevin equations, continuous time random walks and fractional diffusion. Chaos, Solitons & Fractals 2007 34 1 62 80 10.1016/j.chaos.2007.01.050 MR2320428 ZBL1142.82356 (Pubitemid 46541711)
47. Germano G., Politi M., Scalas E., Schilling R. L., Stochastic calculus for uncoupled continuous-time random walks. Physical Review E 2009 79 6 12 066102 10.1103/PhysRevE.79.066102 MR2551283
48. Kiryakova V., virginia@diogenes.bg The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus. Computers and Mathematics with Applications. In press 2-s2.0-70049116485 10.1016/j.camwa.2009.08.025
49. Kiryakova V., virginia@diogenes.bg The special functions of fractional calculus as generalized fractional calculus operators of some basic functions. Computers and Mathematics with Applications 2010 59 3 1128 1141 2-s2.0-67649297800 10.1016/j.camwa.2009.05.014
50. Mainardi F., Gorenflo R., Vivoli A., Renewal processes of Mittag-Leffler and Wright type. Fractional Calculus & Applied Analysis 2005 8 1 7 38 MR2179226 ZBL1126.60070
51. Mainardi F., Pagnini G., The Wright functions as solutions of the time-fractional diffusion equation. Applied Mathematics and Computation 2003 141 1 51 62 10.1016/S0096-3003(02)00320-X MR1984227 ZBL1053.35008
52. Mainardi F., The fundamental solutions for the fractional diffusion-wave equation. Applied Mathematics Letters 1996 9 6 23 28 10.1016/0893-9659(96)00089- 4 MR1419811 ZBL0879.35036 (Pubitemid 126365654)
53. Mainardi F., Luchko Yu., Pagnini G., The fundamental solution of the space-time fractional diffusion equation. Fractional Calculus & Applied Analysis 2001 4 2 153 192 MR1829592 ZBL1054.35156
54. Feller W., An Introduction to Probability Theory and Its Applications. Vol. II 1966 1st New York, NY, USA John Wiley & Sons xviii+636 MR0210154 ZBL0138.10207
55. Feller W., An Introduction to Probability Theory and Its Applications. Vol. II 1971 2nd New York, NY, USA John Wiley & Sons xxiv+669 MR0270403 ZBL0219.60003
56. Humbert P., Nouvelles correspondances symboliques. Bulletin des Sciences Mathématiques 1945 69 121 129 MR0015564 ZBL0060.25112
57. Pollard H., The representation of e-xλ as a Laplace integral. Bulletin of the American Mathematical Society 1946 52 908 910 MR0018286 10.1090/S0002-9904-1946-08672-3 ZBL0060.25007
58. Mikusiński J., On the function whose Laplace-transform is e-sα Studia Mathematica 1959 18 191 198 MR0108694 ZBL0087.10501
59. Buchen P. W., Mainardi F., Asymptotic expansions for transient viscoelastic waves. Journal de Mécanique 1975 14 4 597 608 2-s2.0-0016628745
60. Mainardi F., Gorenflo R., On Mittag-Leffler-type functions in fractional evolution processes. Journal of Computational and Applied Mathematics 2000 118 1-2 283 299 10.1016/S0377-0427(00)00294-6 MR1765955 ZBL0970.45005
61. Achar B. N. N., Hanneken J. W., Clarke T., Damping characteristics of a fractional oscillator. Physica A 2004 339 3-4 311 319 10.1016/j.physa.2004.03. 030 MR2091376
62. Mainardi F., Fractional Calculus and Waves in Linear Viscoelasticity 2010 London, UK Imperial College Press
63. Freed A., Diethelm K., Luchko Yu., Fractional-order viscoelasticity (FOV): constitutive development using the fractional calculus. First Annual Report 2002 NASA/TM-2002-211914 Gleen Research Center
64. Gorenflo R., Loutchko J., Luchko Yu., Computation of the Mittag-Leffler function Eαβ(z) and its derivative. Fractional Calculus & Applied Analysis 2002 5 4 491 518 MR1967847 ZBL1027.33016
65. Podlubny I., Mittag-Leffler function. The MATLAB routine, http://www.mathworks.com/matlabcentral/fileexchange
66. Seybold H. J., Hilfer R., Numerical results for the generalized Mittag-Leffler function. Fractional Calculus & Applied Analysis 2005 8 2 127 139 MR2179658 ZBL1123.33018
67. Doetsch G., Introduction to the Theory and Application of the Laplace Transformation 1974 New York, NY, USA Springer vii+326 MR0344810
68. Mainardi F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons & Fractals 1996 7 9 1461 1477 10.1016/0960-0779(95)00125-5 MR1409912 ZBL1080.26505 (Pubitemid 126377337)
69. Mainardi F., mainardi@bo.infn.it Tomirotti M., Seismic pulse propagation with constant Q and stable probability distributions. Annali di Geofisica 1997 40 5 1311 1328 2-s2.0-0000916799 (Pubitemid 28229916)
70. Luchko Yu., Algorithms for evaluation of the Wright function for the real arguments' values. Fractional Calculus & Applied Analysis 2008 11 1 57 75 MR2379273
71. Mainardi F., Pagnini G., Gorenflo R., Mellin transform and subordination laws in fractional diffusion processes. Fractional Calculus & Applied Analysis 2003 6 4 441 459 MR2044309 ZBL1083.60032
72. Sokolov I. M., Klafter J., From diffusion to anomalous diffusion: a century after Einstein's Brownian motion. Chaos 2005 15 2 7 026103 10.1063/1.1860472 MR2150232 ZBL1080.82022 (Pubitemid 43127627)
73. Mainardi F., Pagnini G., Gorenflo R., Some aspects of fractional diffusion equations of single and distributed order. Applied Mathematics and Computation 2007 187 1 295 305 10.1016/j.amc.2006.08.126 MR2323582 ZBL1122.26004 (Pubitemid 46635728)
74. Saichev A. I., Zaslavsky G. M., Fractional kinetic equations: solutions and applications. Chaos 1997 7 4 753 764 10.1063/1.166272 MR1604710 ZBL0933.37029 (Pubitemid 128005624)
75. Schneider W. R., Wyss W., Fractional diffusion and wave equations. Journal of Mathematical Physics 1989 30 1 134 144 10.1063/1.528578 MR974464 ZBL0692.45004
76. Gel'fand I. M., Shilov G. E., Generalized Functions, Volume I 1964 New York, NY, USA Academic Press
77. Gorenflo R., Mainardi F., Klages R., Radons G., Sokolov I. M., Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects. Anomalous Transport: Foundations and Applications 2008 chapter 4 Weinheim, Germany Wiley-VCH 93 127
78. Meerschaert M. M., Scheffler H.-P., Limit theorems for continuous-time random walks with infinite mean waiting times. Journal of Applied Probability 2004 41 3 623 638 MR2074812 10.1239/jap/1091543414 ZBL1065.60042 (Pubitemid 39620503)
79. Gorenflo R., Mittag-Leffler waiting time, power laws, rarefaction, continuous time random walk, diffusion limit. Workshop on Fractional Calculus and Statistical Distributions November 2009 Pala Campus, Pala-Kerala, India Centre for Mathematical Sciences