Generalized fractional master equation for self-similar stochastic processes modelling anomalous diffusion

International Journal of Stochastic Analysis

Volumn 2012, Issue , 2012, Pages

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

^a CRS4   (Italy)

^b CRESME Research SpA   (Italy)

^c UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84869015438     PISSN: 20903332     EISSN: 20903340     Source Type: Journal    
DOI: 10.1155/2012/427383     Document Type: Article

References (37)

1. Balescu R., Statistical Dynamics. Matter Out of Equilibrium 1997 London, UK Imperial College Press ZBL0997.82505
2. Risken H., The Fokker-Planck Equation: Methods of Solution and Applications 1989 2nd Berlin, Germany Springer 10.1007/978-3-642-61544-3 987631
3. Gardiner C. W., Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences 1990 2nd Berlin, Germany Springer
4. Weiss G. H., Aspects and Applications of the Random Walk 1994 Amsterdam, The Netherlands North-Holland Publishing 1280031
5. Baleanu D., Diethelm K., Scalas E., Trujillo J. J., Fractional Calculus: Models and Numerical Methods 2012 3 Singapore World Scientific Series on Complexity, Nonlinearity and Chaos 10.1142/9789814355216 2894576 ZBL1244.49028
6. Diethelm K., The Analysis of Fractional Differential Equations 2010 Berlin, Germany Springer 10.1007/978-3-642-14574-2 2680847 ZBL1222.37107
7. Metzler R., Klafter J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A 2004 37 31 R161 R208 10.1088/0305-4470/37/31/R01 2090004 ZBL1075.82018 (Pubitemid 39063557)
8. Lundstrom B. N., Higgs M. H., Spain W. J., Fairhall A. L., Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 2008 11 11 1335 1342 2-s2.0-54949148669 10.1038/nn.2212
9. Scalas E., Gorenflo R., Mainardi F., Fractional calculus and continuous-time finance. Physica A 2000 284 1-4 376 384 10.1016/S0378-4371(00) 00255-7 1773804
10. Mainardi F., Raberto M., Gorenflo R., Scalas E., Fractional calculus and continuous-time finance. II: The waiting-time distribution. Physica A 2000 287 3-4 468 481 2-s2.0-0034502929 10.1016/S0378-4371(00)00386-1 (Pubitemid 32034791)
11. Gorenflo R., Mainardi F., Scalas E., Raberto M., Kohlmann M., Tang S., Fractional calculus and continuous-time finance. III. The diffusion limit. Trends in Mathematics-Mathematical Finance 2001 Basel, Switzerland Birkhäuser 171 180
12. Scalas E., The application of continuous-time random walks in finance and economics. Physica A 2006 362 2 225 239 2-s2.0-32544456755 10.1016/j.physa. 2005.11.024 (Pubitemid 43234049)
13. Vinagre B. M., Podlubny I., Hernández A., Feliu V., Some approximations of fractional order operators used in control theory and applications. Fractional Calculus & Applied Analysis 2000 3 3 231 248 1788163 ZBL1111.93302
14. Pagnini G., Nonlinear time-fractional differential equations in combustion science. Fractional Calculus and Applied Analysis 2011 14 1 80 93 10.2478/s13540-011-0006-8 2782246
15. Pagnini G., The evolution equation for the radius of a premixed flame ball in fractional diffusive media. European Physical Journal Special Topics 2011 193 105 117
16. Tenreiro Machado J. A., And I say to myself: What a fractional world! Fractional Calculus and Applied Analysis 2011 14 635 654 10.2478/s13540-011- 0037-1
17. Klafter J., Lim S. C., Metzler R., Fractional Dynamics. Recent Advances 2012 Singapore World Scientific
18. Grigolini P., Rocco A., West B. J., Fractional calculus as a macroscopic manifestation of randomness. Physical Review E 1999 59 3 2603 2613 2-s2.0-0033089733 (Pubitemid 129306019)
19. Rocco A., West B. J., Fractional calculus and the evolution of fractal phenomena. Physica A 1999 265 535 546
20. Schneider W. R., Albeverio S., Casati G., Cattaneo U., Merlini D., Moresi R., Grey noise. Stochastic Processes, Physics and Geometry 1990 Teaneck, NJ, USA World Scientific 676 681
21. Schneider W. R., Albeverio S., Fenstad J. E., Holden H., Lindstrom T., Grey noise. Ideas and Methods in Mathematical Analysis, Stochastics, and Applications 1992 1 Cambridge, UK Cambridge University Press 261 282
22. Mura A., Non-markovian stochastic processes and their applications: from anomalous diffusion to time series analysis [Ph.D. thesis] 2008 University of Bologna. Now available by Lambert Academic Publishing 2011
23. 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 2501791 ZBL1173.26005
24. Mura A., Pagnini G., Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. Journal of Physics A 2008 41 28 10.1088/1751-8113/41/28/285003 285003 2430462 ZBL1143.82028
25. Pagnini G., Erdélyi-Kober fractional diffusion. Fractional Calculus and Applied Analysis 2012 15 1 117 127 10.2478/s13540-012-0008-1 2872114
26. Gorenflo R., Mainardi F., Carpinteri A., Mainardi F., Fractional calculus: integral and differential equations of fractional order. Fractals and Fractional Calculus in Continuum Mechanics 1997 Springer 223 276
27. Mainardi F., Luchko Y., Pagnini G., The fundamental solution of the space-time fractional diffusion equation. Fractional Calculus & Applied Analysis 2001 4 2 153 192 1829592 ZBL1054.35156
28. Mura A., Taqqu M. S., Mainardi F., Non-Markovian diffusion equations and processes: analysis and simulations. Physica A 2008 387 21 5033 5064 10.1016/j.physa.2008.04.035 2588003
29. Mainardi F., Mura A., Pagnini G., The M -Wright function in time-fractional diffusion processes: a tutorial survey. International Journal of Differential Equations 2010 2010 29 104505 2592742
30. Kiryakova V., Generalized Fractional Calculus and Applications 1994 301 Harlow, UK Longman Scientific & Technical 1265940
31. Luchko Y., Operational rules for a mixed operator of the Erdélyi-Kober type. Fractional Calculus & Applied Analysis 2004 7 3 339 364 2252570 ZBL1128.26004
32. Luchko Y., Trujillo J. J., Caputo-type modification of the Erdélyi-Kober fractional derivative. Fractional Calculus & Applied Analysis 2007 10 3 249 267 2382781
33. Mainardi F., Fractional Calculus and Waves in Linear Viscoelasticity 2010 London, UK Imperial College Press 10.1142/9781848163300 2676137
34. Podlubny I., Fractional Differential Equations 1999 198 San Diego, Calif, USA Academic Press 1658022
35. Mainardi F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons and Fractals 1996 7 9 1461 1477 10.1016/0960-0779(95)00125-5 1409912 ZBL1080.26505 (Pubitemid 126377337)
36. Gorenflo R., Luchko Y., Mainardi F., Analytical properties and applications of the Wright function. Fractional Calculus & Applied Analysis 1999 2 4 383 414 1752379 ZBL1027.33006
37. Gorenflo R., Luchko Y., 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 1765948 ZBL0973.35012