Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


IMPACT FACTOR 2018: 3.514
5-year IMPACT FACTOR: 3.524

CiteScore 2018: 3.44

SCImago Journal Rank (SJR) 2018: 1.891
Source Normalized Impact per Paper (SNIP) 2018: 1.808

Mathematical Citation Quotient (MCQ) 2018: 1.08

Online
ISSN
1314-2224
See all formats and pricing
More options …
Volume 19, Issue 2

Issues

A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation

Gianni Pagnini
  • Corresponding author
  • BCAM – Basque Center for Applied Mathematics Alameda de Mazarredo 14, E–48009 Bilbao, Basque Country – SPAIN
  • Ikerbasque - Basque Foundation for Science Calle de Mar´ıa D´ıaz de Haro 3, E–48013 Bilbao, Basque Country – SPAIN
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Paolo Paradisi
  • Corresponding author
  • BCAM – Basque Center for Applied Mathematics Alameda de Mazarredo 14, E–48009 Bilbao, Basque Country – SPAIN
  • ISTI–CNR Istituto di Scienza e Tecnologie dell’ Informazione “A. Faedo” Via Moruzzi 1, I–56124 Pisa, ITALY
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-05-04 | DOI: https://doi.org/10.1515/fca-2016-0022

Abstract

The stochastic solution with Gaussian stationary increments is established for the symmetric space-time fractional diffusion equation when 0 < β < α ≤ 2, where 0 < β ≤ 1 and 0 < α ≤ 2 are the fractional derivation orders in time and space, respectively. This solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with respect to previous approaches such as the scaling limit of the continuous time random walk, the parametric subordination and the subordinated Langevin equation. A new integral representation formula for the fundamental solution of the space-time fractional diffusion equation is firstly derived. It is then shown that, in the symmetric case, a stochastic solution can be obtained by a Gaussian process with stationary increments and with a random wideness scale variable distributed according to an arrangement of two extremal Lévy stable densities. This stochastic solution is self-similar with stationary increments and uniquely defined in a statistical sense by the mean and the covariance structure.

Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental solution of the symmetric space-time fractional diffusion equation.

MSC 2010: Primary 26A33; Secondary 60G20, 60G22, 82C31

Key Words and Phrases: anomalous diffusion; space-time fractional diffusion equation; stochastic solution; Gaussian processes; fractional Brownian motion; self-similar stochastic process; stationary increments

Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 19, No 2 (2016), pp. 408–440, DOI: 10.1515/fca-2016-0022

References

  • [1]

    O.C. Akin, P. Paradisi, P. Grigolini, Perturbation-induced emergence of poisson-like behavior in non-poisson systems. J. Stat. Mech.: Theory Exp. (2009), P01013.Google Scholar

  • [2]

    O.C. Akin, P. Paradisi, P. Grigolini, Periodic trend and fluctuations: The case of strong correlation. Physica A 371 (2006), 157–170.Google Scholar

  • [3]

    P. Allegrini, P. Paradisi, D. Menicucci, M. Laurino, R. Bedini, A. Piarulli, A. Gemignani, Sleep unconsciousness and breakdown of serial critical intermittency: New vistas on the global workspace. Chaos Soli-tons Fract. 55 (2013), 32–43.Google Scholar

  • [4]

    B. Baeumer, M.M. Meerschaert, Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4 (2001), 481–500.Google Scholar

  • [5]

    B. Baeumer, M.M. Meerschaert, E. Nane, Brownian subordinators and fractional Cauchy problems. Trans. Amer. Math. Soc. 361, No 7 (2009), 3915–3930.Google Scholar

  • [6]

    B. Baeumer, M.M. Meerschaert, E. Nane, Space-time fractional diffusion. J. Appl. Prob. 46 (2009), 1100–1115.Google Scholar

  • [7]

    D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos Vol. 3, World Sci. Publ., New Jersey (2012).Google Scholar

  • [8]

    E. Barkai, CTRW pathways to the fractional diffusion equation. Chem. Phys. 284 (2002), 13–27.Google Scholar

  • [9]

    D. A. Benson, M.M. Meerschaert, J. Revielle, Fractional calculus in hydrologic modeling: A numerical perspective. Adv. Water Resour. 51 (2013), 479–497.Google Scholar

  • [10]

    F. Biagini, Y. Hu, B. Øksendal, T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications. Springer (2008).Google Scholar

  • [11]

    S. Bianco, P. Grigolini, P. Paradisi, A fluctuating environment as a source of periodic modulation. Chem. Phys. Lett. 438, No 4-6 (2007), 336–340.Google Scholar

  • [12]

    D. O. Cahoy, On the parametrization of the M-Wright function. Far East J. Theor. Stat. 34, No 2 (2011), 155–164.Google Scholar

  • [13]

    D.O. Cahoy, Estimation and simulation for the M-Wright function. Commun. Stat.-Theor. M. 41, No 8 (2012), 1466–1477.Google Scholar

  • [14]

    D.O. Cahoy, Moment estimators for the two-parameter M-Wright distribution. Computation. Stat. 27, No 3 (2012), 487–497.Google Scholar

  • [15]

    P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, A. Vulpiani, On strong anomalous diffusion. Physica D 134 (1999), 75–93.Google Scholar

  • [16]

    J.M. Chambers, C.L. Mallows, B.W. Stuck, A method for simulating skewed stable random variables. J. Amer. Statist. Assoc. 71, (1976), 340–344.Google Scholar

  • [17]

    M. Chevrollier, N. Mercadier, W. Guerin, R. Kaiser, Anomalous photon diffusion in atomic vapors. Eur. Phys. J. D 58 (2010), 161–165.Google Scholar

  • [18]

    A. Compte, Stochastic foundations of fractional dynamics. Phys. Rev. E 53, No 4 (1996), 4191–4193.Google Scholar

  • [19]

    D.R. Cox, Renewal Theory. Methuen & Co. Ltd., London (1962).Google Scholar

  • [20]

    J.L. da Silva, Local times for grey Brownian motion. Int. J. Mod. Phys. Conf. Ser. 36 (2015), 1560003. [7th Jagna Int. Workshop (2014)].Google Scholar

  • [21]

    J.L. da Silva, M. Erraoui, Grey Brownian motion local time: Existence and weak-approximation. Stochastics 87 (2014), 347–361.Google Scholar

  • [22]

    D. del Castillo-Negrete, Fractional diffusion in plasma turbulence. Phys. Plasmas 11, No 8 (2004), 3854–3864.Google Scholar

  • [23]

    D. del Castillo-Negrete, Non-diffusive, non-local transport in fluids and plasmas. Nonlin. Processes Geophys. 17 (2010), 795–807.Google Scholar

  • [24]

    D. del Castillo-Negrete, B.A. Carreras, V.E. Lynch, Nondiffusive transport in plasma turbulence: A fractional diffusion approach. Phys. Rev. Lett. 94 (2005), 065003.Google Scholar

  • [25]

    D. del Castillo-Negrete, P. Mantica, V. Naulin, J.J. Rasmussen, JET EFDA contributors, Fractional diffusion models of non-local perturbative transport: numerical results and application to JET experiments. Nucl. Fusion 48 (2008), 075009.Google Scholar

  • [26]

    T. Dieker, Simulation of Fractional Brownian Motion. Ph.D. Thesis, CWI and University of Twente, The Netherlands (2004).Google Scholar

  • [27]

    P. Dieterich, R. Klages, R. Preuss, A. Schwab, Anomalous dynamics of cell migration. Proc. Nat. Acad. Sci. 105, No 2 (2008), 459–463.Google Scholar

  • [28]

    G. Dif-Pradalier, P.H. Diamond, V. Grandgirard, Y. Sarazin, J. Abiteboul, X. Garbet, Ph. Ghendrih, A. Strugarek, S. Ku, C. S. Chang, On the validity of the local diffusive paradigm in turbulent plasma transport. Phys. Rev. E 82 (2010), 025401(R).Google Scholar

  • [29]

    B. Dybiec, Anomalous diffusion: temporal non-Markovianity and weak ergodicity breaking. J. Stat. Mech.-Theory Exp. (2009), P08025.Google Scholar

  • [30]

    B. Dybiec, E. Gudowska-Nowak, Subordinated diffusion and continuous time random walk asymptotics. Chaos 20, No 4 (2010), 043129.Google Scholar

  • [31]

    S. Eule, R. Friedrich, Subordinated Langevin equations for anomalous diffusion in external potentials–biasing and decoupled external forces. Europhys. Lett. 86 (2009), 3008.Google Scholar

  • [32]

    W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, 2nd Ed. Wiley, New York (1971).Google Scholar

  • [33]

    H.C. Fogedby, Langevin equations for continuous time Lévy flights. Phys. Rev. E 50, No 2 (1994), 1657–1660.Google Scholar

  • [34]

    D. Fulger, E. Scalas, G. Germano, Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Phys. Rev. E 77 (2008), 021122.Google Scholar

  • [35]

    D. Fulger, E. Scalas, G. Germano, Random numbers form the tails of probability distributions using the transformation method. Fract. Calc. Appl. Anal. 16, No 2 (2013), 332–353; DOI: 10.2478/s13540-013-0021-z; http://www.degruyter.com/view/j/fca.2013.16.issue-2/issue-files/fca.2013.16.issue-2.xml.Crossref

  • [36]

    G. Germano, M. Politi, E. Scalas, R. L. Schilling, Stochastic calculus for uncoupled continuous-time random walks. Phys. Rev. E 79, No 6 (2009), 066102.Google Scholar

  • [37]

    R. Gorenflo, A. Iskenderov, Yu. Luchko, Mapping between solutions of fractional diffusion-wave equations. Fract. Calc. Appl. Anal. 3, No 1 (2000), 75–86.Google Scholar

  • [38]

    R. Gorenflo, F. Mainardi, Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal. 1, No 2 (1998), 167–191.Google Scholar

  • [39]

    R. Gorenflo, F. Mainardi, Some recent advances in theory and simulation of fractional diffusion processes. J. Comput. Appl. Math. 229, No 2 (2009), 400–415.Google Scholar

  • [40]

    R. Gorenflo, F. Mainardi, Subordination pathways to fractional diffusion. Eur. Phys. J. Special Topics 193 (2011), 119–132.Google Scholar

  • [41]

    R. Gorenflo, F. Mainardi, Parametric subordination in fractional diffusion processes. In: J. Klafter, S.C. Lim, and R. Metzler (Ed-s), Fractional Dynamics. Recent Advances, World Sci., Singapore (2012), 227–261.Google Scholar

  • [42]

    R. Gorenflo, F. Mainardi, D. Moretti, G. Pagnini, P. Paradisi, Discrete random walk models for space-time fractional diffusion. Chem. Phys. 284 (2002), 521–541.Google Scholar

  • [43]

    R. Gorenflo, F. Mainardi, D. Moretti, G. Pagnini, P. Paradisi, Fractional diffusion: probability distributions and random walk models. Physica A 305, No 1-2 (2002), 106–112.Google Scholar

  • [44]

    R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Time fractional diffusion: A discrete random walk approach. Nonlinear Dynam. 29, No 1-4 (2002), 129–143.Google Scholar

  • [45]

    R. Gorenflo, F. Mainardi, A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion. Chaos Solitons Fract. 34, No 1 (2007), 87–103.Google Scholar

  • [46]

    P. Grigolini, A. Rocco, B. J. West, Fractional calculus as a macroscopic manifestation of randomness. Phys. Rev. E 59, No 3 (1999), 2603–2613.Google Scholar

  • [47]

    K. Gustafson, D. del Castillo-Negrete, W. Dorland, Finite Larmor radius effects on nondiffusive tracer transport in zonal flows. Phys. Plasmas 15 (2008), 102309.Google Scholar

  • [48]

    J. Honkonen, Stochastic processes with stable distributions in random environments. Phys. Rev. E 55, No 1 (1996), 327–331.Google Scholar

  • [49]

    J.R.M. Hosking, Modeling persistence in hydrological time series using fractional differencing. Water Resour. Res. 20 (1984), 1898–1908.Google Scholar

  • [50]

    B.D. Hughes, Anomalous diffusion, stable processes, and generalized functions. Phys. Rev. E 65 (2002), 035105(R).Google Scholar

  • [51]

    J. Klafter, I.M. Sokolov, Anomalous diffusion spread its wings. Physics World 18 (2005), 29–32.Google Scholar

  • [52]

    D. Kleinhans, R. Friedrich, Continuous-time random walks: Simulation of continuous trajectories. Phys. Rev. E 76 (2007), 061102.Google Scholar

  • [53]

    X. Leoncini, L. Kuznetsov, G. Zaslavsky, Evidence of fractional transport in point vortex flow. Chaos Solitons Fract. 19 (2004), 259–273.Google Scholar

  • [54]

    Yu. Luchko, Fractional wave equation and damped waves. J. Math. Phys. 54 (2013), 031505.Google Scholar

  • [55]

    M. Magdziarz, A. Weron, J. Klafter, Equivalence of the fractional Fokker–Planck and subordinated Langevin equations: The case of a time-dependent force. Phys. Rev. Lett. 101 (2008), 210601.Google Scholar

  • [56]

    F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fract. 7 (1996), 1461–1477.Google Scholar

  • [57]

    F. Mainardi, Yu. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, No 2 (2001), 153–192.Google Scholar

  • [58]

    F. Mainardi, A. Mura, G. Pagnini, The functions of the Wright type in fractional calculus. Lecture Notes of Seminario Interdisciplinare di Matematica 9 (2010), 111–128.Google Scholar

  • [59]

    F. Mainardi, A. Mura, G. Pagnini, The M-Wright function in time-fractional diffusion processes: A tutorial survey. Int. J. Differ. Equations 2010 (2010), 104505.Google Scholar

  • [60]

    F. Mainardi, G. Pagnini, The Wright functions as solutions of the time-fractional diffusion equations. Appl. Math. Comput. 141 (2003), 51–62.Google Scholar

  • [61]

    F. Mainardi, G. Pagnini, R. Gorenflo, Mellin transform and subordination laws in fractional diffusion processes. Fract. Calc. Appl. Anal. 6, No 4 (2003), 441–459.Google Scholar

  • [62]

    F. Mainardi, G. Pagnini, R. Gorenflo, Mellin convolution for subordinated stable processes. J. Math. Sci. 132, No 5 (2006), 637–642.Google Scholar

  • [63]

    F. Mainardi, G. Pagnini, R. K. Saxena, Fox H functions in fractional diffusion. J. Comput. Appl. Math. 178 (2005), 321–331.Google Scholar

  • [64]

    M.M. Meerschaert, D. A. Benson, H.-P. Scheffler, B. Baeumer, Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65 (2002), 041103.Google Scholar

  • [65]

    M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus. De Gruyter (2012).Google Scholar

  • [66]

    Y. Meroz, I.M. Sokolov, J. Klafter, Unequal twins: Probability distributions do not determine everything. Phys. Rev. Lett. 107 (2011), 260601.Google Scholar

  • [67]

    R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in fractional dynamics descriptions of anomalous dynamical processes. J. Phys. A: Math. Theor. 37, No 31 (2004), R161–R208.Google Scholar

  • [68]

    R. Metzler, T.F. Nonnenmacher, Space- and time-fractional diffusion and wave equations, fractional Fokker–Planck equations, and physical motivation. Chem. Phys. 284 (2002), 67–90.Google Scholar

  • [69]

    E.W. Montroll, Random walks on lattices. Proc. Symp. Appl. Math. Am. Math. Soc. 16 (1964), 193–220.Google Scholar

  • [70]

    E.W. Montroll, G.H. Weiss, Random walks on lattices, II. J. Math. Phys. 6 (1965), 167–181.Google Scholar

  • [71]

    A. Mura, Non-Markovian Stochastic Processes and Their Applications: From Anomalous Diffusion to Time Series Analysis. Ph.D. Thesis, Physics Dept., Univ. of Bologna (2008); Lambert Acad. Publ. (2011).Google Scholar

  • [72]

    A. Mura, F. Mainardi, A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics. Integr. Transf. Spec. Funct. 20, No 3-4 (2009), 185–198.Google Scholar

  • [73]

    A. Mura, G. Pagnini, Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. J. Phys. A: Math. Theor. 41 (2008), 285003.Google Scholar

  • [74]

    G. Pagnini, Erdélyi–Kober fractional diffusion. Fract. Calc. Appl. Anal. 15, No 1 (2012), 117–127; DOI: 10.2478/s13540-012-0008-1; http://www.degruyter.com/view/j/fca.2012.15.issue-1/issue-files/fca.2012.15.issue-1.xml.Crossref

  • [75]

    G. Pagnini, The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes. Fract. Calc. Appl. Anal. 16, No 2 (2013), 436–453; DOI: 10.2478/s13540-013-0027-6; http://www.degruyter.com/view/j/fca.2013.16.issue-2/issue-files/fca.2013.16.issue-2.xml.Crossref

  • [76]

    G. Pagnini, Self-similar stochastic models with stationary increments for symmetric space-time fractional diffusion. In:Proc. 10th IEEE/ASME Internat. Conf. on Mechatronic and Embedded Systems and Applications, MESA 2014, Senigallia (AN), Italy, 10–12 Sept. (2014), Paper Code MESA2014 003; doi:10.1109/MESA.2014.6935520.CrossrefGoogle Scholar

  • [77]

    G. Pagnini, Short note on the emergence of fractional kinetics. Physica A 409 (2014), 29–34.Google Scholar

  • [78]

    G. Pagnini, Subordination formulae for space-time fractional diffusion processes via Mellin convolution. In: P.M. Pardalos, R.P. Agarwal, L. Kočcinac, R. Neck, N. Mastorakis, K. Ntalianas (Ed-s), Recent Advances in Mathematics, Statistics and Economics. Proc. of the 2014 Internat. Conf. on Pure Mathematics –Applied Mathematics (PMAM’14), Venice, Italy, 15–17 March (2014), 40–45.Google Scholar

  • [79]

    G. Pagnini, Generalized Equations for Anomalous Diffusion and Their Fundamental Solutions. Thesis for Degree in Physics, Univ. of Bologna (Oct. 2000), In Italian.Google Scholar

  • [80]

    G. Pagnini, A. Mura, F. Mainardi, Generalized fractional master equation for self-similar stochastic processes modelling anomalous diffusion. Int. J. Stoch. Anal. 2012 (2012), 427383.Google Scholar

  • [81]

    G. Pagnini, A. Mura, F. Mainardi, Two-particle anomalous diffusion: Probability density functions and self-similar stochastic processes. Phil. Trans. R. Soc. A 371 (2013), 20120154.Google Scholar

  • [82]

    G. Pagnini, E. Scalas, Historical notes on the M-Wright/Mainardi function. Commun. in Appl. and Industr. Math. 6, No 1 (2014), e–496; DOI: 10.1685/journal.caim.496 (Editorial).CrossrefGoogle Scholar

  • [83]

    P. Paradisi, Fractional calculus in statistical physics: The case of time fractional diffusion equation. Commun. in Appl. and Industr. Math. 6, No 2 (2014), e–530; doi: 10.1685/journal.caim.530.CrossrefGoogle Scholar

  • [84]

    P. Paradisi, P. Allegrini, A. Gemignani, M. Laurino, D. Menicucci, A. Piarulli, Scaling and intermittency of brain events as a manifestation of consciousness. AIP Conf. Proc. 1510 (2013), 151–161.Google Scholar

  • [85]

    P. Paradisi, R. Cesari, D. Contini, A. Donateo, L. Palatella, Characterizing memory in atmospheric time series: an alternative approach based on renewal theory. Eur. Phys. J. Special Topics 174 (2009), 207–218.Google Scholar

  • [86]

    P. Paradisi, R. Cesari, A. Donateo, D. Contini, P. Allegrini, Diffusion scaling in event-driven random walks: an application to turbulence. Rep. Math. Phys. 70 (2012), 205–220.Google Scholar

  • [87]

    P. Paradisi, R. Cesari, A. Donateo, D. Contini, P. Allegrini, Scaling laws of diffusion and time intermittency generated by coherent structures in atmospheric turbulence. Nonlin. Processes Geophys. 19 (2012), 113–126; Corrigendum, in:Nonlin. Processes Geophys. 19 (2012), 685.Google Scholar

  • [88]

    P. Paradisi, R. Cesari, P. Grigolini, Superstatistics and renewal critical events. Cent. Eur. J. Phys. 7 (2009), 421–431.Google Scholar

  • [89]

    P. Paradisi, R. Cesari, F. Mainardi, A. Maurizi, F. Tampieri, A generalized Fick’s law to describe non-local transport effects. Phys. Chem. Earth 26, No 4 (2001), 275–279.Google Scholar

  • [90]

    P. Paradisi, R. Cesari, F. Mainardi, F. Tampieri, The fractional Fick’s law for non-local transport processes. Physica A 293, No 1-2 (2001), 130–142.Google Scholar

  • [91]

    P. Paradisi, D. Chiarugi, P. Allegrini, A renewal model for the emergence of anomalous solute crowding in liposomes. BMC Syst. Biol. 9, Suppl. 3 (2015), s7.Google Scholar

  • [92]

    I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Google Scholar

  • [93]

    S. Ratynskaia, K. Rypdal, C. Knapek, S. Khrapak, A.V. Milovanov, A. Ivlev, J.J. Rasmussen, G.E. Morfill, Superdiffusion and viscoelastic vortex flows in a two-dimensional complex plasma. Phys. Rev. Lett. 96, No 10 (2006), 105010.Google Scholar

  • [94]

    A. Rocco, B.J. West, Fractional calculus and the evolution of fractal phenomena. Physica A 265, No 3-4 (1999), 535–546.Google Scholar

  • [95]

    A. Saichev, G. Zaslavsky, Fractional kinetic equations: solutions and applications. Chaos 7 (1997), 753–764.Google Scholar

  • [96]

    E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance. Physica A 284 (2000), 376–384.Google Scholar

  • [97]

    E. Scalas, R. Gorenflo, F. Mainardi, Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. Phys. Rev. E 69 (2004), 011107.Google Scholar

  • [98]

    M. Schmiedeberg, V.Yu. Zaburdaev, H. Stark, On moments and scaling regimes in anomalous random walks. J. Stat. Mech.-Theory Exp. (2009), P12020.Google Scholar

  • [99]

    W.R. Schneider, W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30, No 1 (1989), 134–144.Google Scholar

  • [100]

    J.H.P. Schulz, A.V. Chechkin, R. Metzler, Correlated continuos time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics. J. Phys. A: Math. Theor. 46 (2013), 475001.Google Scholar

  • [101]

    I.M. Sokolov, J. Klafter, A. Blumen, Fractional kinetics. Physics Today 55 (2002), 48–54.Google Scholar

  • [102]

    I.M. Sokolov, R. Metzler, Non-uniqueness of the first passage time density of Lévy random processes. J. Phys. A: Math. Theor. 37, (2004), L609–L615.Google Scholar

  • [103]

    V.V. Uchaikin, Montroll–Weiss problem, fractional equations and stable distributions. Int. J. Theor. Phys. 39 (2000), 2087–2105.Google Scholar

  • [104]

    V.V. Uchaikin, V.M. Zolotarev, Chance and Stability. Stable Distributions and their Applications. VSP, Utrecht (1999).Google Scholar

  • [105]

    G.H. Weiss, R.J. Rubin, Random walks: Theory and selected applications. Adv. Chem. Phys. 52 (1983), 363–505.Google Scholar

  • [106]

    A. Weron, M. Magdziarz, K. Weron, Modeling of subdiffusion in space-time-dependent force fields beyond the fractional Fokker–Planck equation. Phys. Rev. E 77 (2008), 036704.Google Scholar

  • [107]

    R. Weron, On the Chambers–Mallows–Stuck method for simulating skewed stable random variables. Statist. Probab. Lett. 28 (1996), 165–171; Corrigendum: http://mpra.ub.uni-muenchen.de/20761/1/RWeron96 Corr.pdf or http://www.im.pwr.wroc.pl/∼ hugo/ RePEc/wuu/wpaper/HSC 96 01.pdf.

  • [108]

    G.M. Zaslavsky, Anomalous transport and fractal kinetics. In: H.K. Moffatt, G.M. Zaslavsky, P. Compte, and M. Tabor (Ed-s), Topological Aspects of the Dynamics of Fluids and Plasmas, NATO ASI Series Vol. 218, Kluwer, Dordrecht (1992), 481–491.Google Scholar

  • [109]

    G.M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos. Physica D 76 (1994), 110–122.Google Scholar

  • [110]

    G.M. Zaslavsky, Renormalization group theory of anomalous transport in systems with Hamiltonian chaos. Chaos 4 (1994), 25–33.Google Scholar

  • [111]

    G.M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, (2002), 461–580.Google Scholar

  • [112]

    G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005).Google Scholar

  • [113]

    G.M. Zaslavsky, B.A. Niyazov, Fractional kinetics and accelerator modes. Phys. Rep. 283 (1997), 73–93.Google Scholar

About the article

Received: 2015-05-05

Revised: 2015-11-30

Published Online: 2016-05-04

Published in Print: 2016-04-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 19, Issue 2, Pages 408–440, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2016-0022.

Export Citation

© 2016 Diogenes Co., Sofia.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Silvia Vitali, Iva Budimir, Claudio Runfola, and Gastone Castellani
Mathematics, 2019, Volume 7, Number 12, Page 1145
[2]
Francesco Di Tullio, Paolo Paradisi, Renato Spigler, and Gianni Pagnini
Frontiers in Physics, 2019, Volume 7
[3]
É. C. Rocha, M. G. E. da Luz, E. P. Raposo, and G. M. Viswanathan
Physical Review E, 2019, Volume 100, Number 1
[4]
Фатима Гидовна Хуштова and Fatima Gidovna Khushtova
Вестник Самарского государственного технического университета. Серия «Физико-математические науки», 2018, Volume 22, Number 4, Page 774
[5]
Oleksii Yu Sliusarenko, Silvia Vitali, Vittoria Sposini, Paolo Paradisi, Aleksei Chechkin, Gastone Castellani, and Gianni Pagnini
Journal of Physics A: Mathematical and Theoretical, 2019, Volume 52, Number 9, Page 095601
[6]
Trifce Sandev, Weihua Deng, and Pengbo Xu
Journal of Physics A: Mathematical and Theoretical, 2018, Volume 51, Number 40, Page 405002
[7]
Silvia Vitali, Vittoria Sposini, Oleksii Sliusarenko, Paolo Paradisi, Gastone Castellani, and Gianni Pagnini
Journal of The Royal Society Interface, 2018, Volume 15, Number 145, Page 20180282
[8]
Vittoria Sposini, Aleksei V Chechkin, Flavio Seno, Gianni Pagnini, and Ralf Metzler
New Journal of Physics, 2018, Volume 20, Number 4, Page 043044
[9]
Daniel Molina-García, Tuan Minh Pham, Paolo Paradisi, Carlo Manzo, and Gianni Pagnini
Physical Review E, 2016, Volume 94, Number 5

Comments (0)

Please log in or register to comment.
Log in