Fractional diffusion: probability distributions and random walk models

Gorenflo, Rudolf; Mainardi, Francesco; Moretti, Daniele; Pagnini, Gianni; Paradisi, Paolo

doi:10.1016/s0378-4371(01)00647-1

Cited by 82 publications

(51 citation statements)

References 20 publications

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“…which, when the medium is homogeneous, takes the form of the well-known Fractional Diffusion Equation (FDE) with Kilbas et al, 2006;Podlubny, 1999;Oldham, Spanier, 1974;Miller, Ross, 1993;Mainardi et al, 2005;Gorenflo et al, 2004;Gorenflo et al, 2002;Metzler, Klafter, 2000a;Metzler, Klafter, 2004). Moreover, we remark that the inner x ∂ ∂ in (37) operates on a quantity which has the dimensions of the product of a speed times a concentration, similarly as in the case of Fickian diffusion.…”

Section: Analytical Formulation Of the Problemmentioning

confidence: 95%

Normal and anomalous transport across an interface: Monte Carlo and analytical approach

Marseguerra

Zoia

2006

Annals of Nuclear Energy

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We present a Monte Carlo scheme to simulate particles going across an interface separating two layers of a medium characterized by different physical properties, together with an analytical formulation of the same problem, for both normal diffusive and subdiffusive regimes. We relate the Monte Carlo simulation parameters to the coefficients and boundary conditions appearing in the companion analytical equations. Under suitable physical hypotheses on the constraints to be imposed on such parameters, we show that the Monte Carlo simulation results are in good agreement with the corresponding analytical solutions. In particular, we remark that -while in the normal diffusion case the conservation of particle local velocities across the interface leads to a smoothly varying concentration profile -in the subdiffusive case the same condition leads to a neat jump in resident concentration.

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Section: Analytical Formulation Of the Problemmentioning

confidence: 95%

Normal and anomalous transport across an interface: Monte Carlo and analytical approach

Marseguerra

Zoia

2006

Annals of Nuclear Energy

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“…Firstly, the idea of this area appeared in a letter by Leibniz to L' Hospital in 17 th century, Podlubny (1999) . N. Sneddon, S. G. Samko, T. J. Osler, E. R. Love, and many others Boumenir and Tuan, (2010a and b), Chechkin et al (2003), Freiling and Yurko (2001) and Gorenflo et al (2002). Fractional diffusion equations have been investigated in a lot of different physical situations.…”

Section: Introductionmentioning

confidence: 99%

<b>The Inverse Nodal problem for the fractional diffusion equation

Baş

2015

Acta Sci. Technol.

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“…A successful one is that based on evolution equations of fractional order and the name fractional diffusion follows. This approach has been maintained by analytical methods, e.g., [6,7,37], by random walks, e.g., [15,20,21,22], by the continuous time random walk (CTRW) through a well-scaled passage to the diffusion limit, e.g., [16,17,18,23], by the parametric subordination, e.g., [19], and by the inverse stable subordination, e.g., [1,42].…”

Section: Introductionmentioning

confidence: 99%

The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes

Pagnini

2013

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The leading role of a special function of the Wright-type, referred to as M-Wright or Mainardi function, within a parametric class of self-similar stochastic processes with stationary increments, is surveyed. This class of processes, known as generalized grey Brownian motion, provides models for both fast and slow anomalous diffusion. In view of a subordination-type formula involving M-Wright functions, these processes emerge to have all finite moments and be uniquely defined by their mean and auto-covariance structure like Gaussian processes. The corresponding master equation is shown to be a fractional differential equation in the Erdélyi-Kober sense and the diffusive process is named Erdélyi-Kober fractional diffusion. In Appendix, an historical overview on the M-Wright function is reported.MSC 2010 : Primary 33E20; Secondary 26A33, 33E30, 44A35, 60G18, 60G22

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Fractional diffusion: probability distributions and random walk models

Cited by 82 publications

References 20 publications

Normal and anomalous transport across an interface: Monte Carlo and analytical approach

Normal and anomalous transport across an interface: Monte Carlo and analytical approach

<b>The Inverse Nodal problem for the fractional diffusion equation

The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes

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