zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Generalized fractional master equation for self-similar stochastic processes modelling anomalous diffusion. (English) Zbl 1260.60163
Summary: The master equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting master equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function) emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

MSC:
60J60Diffusion processes
60G18Self-similar processes
60G22Fractional processes, including fractional Brownian motion
WorldCat.org
Full Text: DOI
References:
[1] R. Balescu, Statistical Dynamics. Matter Out of Equilibrium, Imperial College Press, London, UK, 1997. · Zbl 0997.82505
[2] H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Springer, Berlin, Germany, 2nd edition, 1989. · Zbl 0665.60084 · doi:10.1007/978-3-642-61544-3
[3] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer, Berlin, Germany, 2nd edition, 1990. · Zbl 0713.60076
[4] G. H. Weiss, Aspects and Applications of the Random Walk, North-Holland Publishing, Amsterdam, The Netherlands, 1994. · Zbl 0925.60079
[5] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific, Singapore, 2012. · Zbl 1244.49028 · doi:10.1016/j.amc.2012.02.080
[6] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010. · Zbl 1222.37107 · doi:10.1016/j.cnsns.2010.01.011
[7] R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A, vol. 37, no. 31, pp. R161-R208, 2004. · Zbl 1075.82018 · doi:10.1088/0305-4470/37/31/R01
[8] B. N. Lundstrom, M. H. Higgs, W. J. Spain, and A. L. Fairhall, “Fractional differentiation by neocortical pyramidal neurons,” Nature Neuroscience, vol. 11, no. 11, pp. 1335-1342, 2008. · doi:10.1038/nn.2212
[9] E. Scalas, R. Gorenflo, and F. Mainardi, “Fractional calculus and continuous-time finance,” Physica A, vol. 284, no. 1-4, pp. 376-384, 2000. · doi:10.1016/S0378-4371(00)00255-7
[10] F. Mainardi, M. Raberto, R. Gorenflo, and E. Scalas, “Fractional calculus and continuous-time finance. II: The waiting-time distribution,” Physica A, vol. 287, no. 3-4, pp. 468-481, 2000. · Zbl 1138.91444 · doi:10.1016/S0378-4371(00)00386-1
[11] R. Gorenflo, F. Mainardi, E. Scalas, and M. Raberto, “Fractional calculus and continuous-time finance. III. The diffusion limit,” in Trends in Mathematics-Mathematical Finance, M. Kohlmann and S. Tang, Eds., pp. 171-180, Birkhäuser, Basel, Switzerland, 2001. · Zbl 1138.91444
[12] E. Scalas, “The application of continuous-time random walks in finance and economics,” Physica A, vol. 362, no. 2, pp. 225-239, 2006. · doi:10.1016/j.physa.2005.11.024
[13] B. M. Vinagre, I. Podlubny, A. Hernández, and V. Feliu, “Some approximations of fractional order operators used in control theory and applications,” Fractional Calculus & Applied Analysis, vol. 3, no. 3, pp. 231-248, 2000. · Zbl 1111.93302
[14] G. Pagnini, “Nonlinear time-fractional differential equations in combustion science,” Fractional Calculus and Applied Analysis, vol. 14, no. 1, pp. 80-93, 2011. · Zbl 1273.34013 · doi:10.2478/s13540-011-0006-8
[15] G. Pagnini, “The evolution equation for the radius of a premixed flame ball in fractional diffusive media,” in European Physical Journal Special Topics, vol. 193, pp. 105-117, 2011.
[16] J. A. Tenreiro Machado, “And I say to myself: “What a fractional world!”,” Fractional Calculus and Applied Analysis, vol. 14, pp. 635-654, 2011. · Zbl 1273.37002 · doi:10.2478/s13540-011-0037-1
[17] J. Klafter, S. C. Lim, and R. Metzler, Eds., Fractional Dynamics. Recent Advances, World Scientific, Singapore, 2012. · Zbl 1238.93005
[18] P. Grigolini, A. Rocco, and B. J. West, “Fractional calculus as a macroscopic manifestation of randomness,” Physical Review E, vol. 59, no. 3, pp. 2603-2613, 1999.
[19] A. Rocco and B. J. West, “Fractional calculus and the evolution of fractal phenomena,” Physica A, vol. 265, pp. 535-546, 1999.
[20] W. R. Schneider, “Grey noise,” in Stochastic Processes, Physics and Geometry, S. Albeverio, G. Casati, U. Cattaneo, D. Merlini, and R. Moresi, Eds., pp. 676-681, World Scientific, Teaneck, NJ, USA, 1990.
[21] W. R. Schneider, “Grey noise,” in Ideas and Methods in Mathematical Analysis, Stochastics, and Applications, S. Albeverio, J. E. Fenstad, H. Holden, and T. Lindstrøm, Eds., vol. 1, pp. 261-282, Cambridge University Press, Cambridge, UK, 1992. · Zbl 0761.60036
[22] A. Mura, Non-markovian stochastic processes and their applications: from anomalous diffusion to time series analysis [Ph.D. thesis], University of Bologna, 2008, Now available by Lambert Academic Publishing 2011.
[23] A. Mura and F. Mainardi, “A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics,” Integral Transforms and Special Functions, vol. 20, no. 3-4, pp. 185-198, 2009. · Zbl 1173.26005 · doi:10.1080/10652460802567517
[24] A. Mura and G. Pagnini, “Characterizations and simulations of a class of stochastic processes to model anomalous diffusion,” Journal of Physics A, vol. 41, no. 28, Article ID 285003, 2008. · Zbl 1143.82028 · doi:10.1088/1751-8113/41/28/285003
[25] G. Pagnini, “Erdélyi-Kober fractional diffusion,” Fractional Calculus and Applied Analysis, vol. 15, no. 1, pp. 117-127, 2012. · Zbl 1276.26021 · doi:10.2478/s13540-012-0008-1
[26] R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., pp. 223-276, Springer, 1997.
[27] F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153-192, 2001. · Zbl 1054.35156
[28] A. Mura, M. S. Taqqu, and F. Mainardi, “Non-Markovian diffusion equations and processes: analysis and simulations,” Physica A, vol. 387, no. 21, pp. 5033-5064, 2008. · doi:10.1016/j.physa.2008.04.035
[29] F. Mainardi, A. Mura, and G. Pagnini, “The M-Wright function in time-fractional diffusion processes: a tutorial survey,” International Journal of Differential Equations, vol. 2010, Article ID 104505, 29 pages, 2010. · Zbl 1222.60060 · doi:10.1155/2010/104505 · eudml:233329
[30] V. Kiryakova, Generalized Fractional Calculus and Applications, vol. 301, Longman Scientific & Technical, Harlow, UK, 1994. · Zbl 0882.26003
[31] Y. Luchko, “Operational rules for a mixed operator of the Erdélyi-Kober type,” Fractional Calculus & Applied Analysis, vol. 7, no. 3, pp. 339-364, 2004. · Zbl 1128.26004 · eudml:11251
[32] Y. Luchko and J. J. Trujillo, “Caputo-type modification of the Erdélyi-Kober fractional derivative,” Fractional Calculus & Applied Analysis, vol. 10, no. 3, pp. 249-267, 2007. · Zbl 1152.26304 · eudml:11329
[33] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010. · Zbl 1210.26004 · doi:10.1142/9781848163300
[34] I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[35] F. Mainardi, “Fractional relaxation-oscillation and fractional diffusion-wave phenomena,” Chaos, Solitons and Fractals, vol. 7, no. 9, pp. 1461-1477, 1996. · Zbl 1080.26505 · doi:10.1016/0960-0779(95)00125-5
[36] R. Gorenflo, Y. Luchko, and F. Mainardi, “Analytical properties and applications of the Wright function,” Fractional Calculus & Applied Analysis, vol. 2, no. 4, pp. 383-414, 1999. · Zbl 1027.33006
[37] R. Gorenflo, Y. Luchko, and F. Mainardi, “Wright functions as scale-invariant solutions of the diffusion-wave equation,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 175-191, 2000. · Zbl 0973.35012 · doi:10.1016/S0377-0427(00)00288-0