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- Francesco Mainardi, Antonio Mura, Gianni Pagnini, Rudolf Gorenflo
- 2008

The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general distribution of time orders we provide the fundamental solution, that is still a probability density, in terms of an integral of… (More)

- Francesco Mainardi, Gianni Pagnini
- 2008

The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin-Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox-Wright functions. A series expansion is also provided in order to point… (More)

The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order β ∈ (0, 1). The fundamental solution for the Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process related to a phenomenon of… (More)

- Francesco Mainardi, Gianni Pagnini, Rudolf Gorenflo
- 2003

The Mellin transform is usually applied in probability theory to the product of independent random variables. In recent times the machinery of the Mellin transform has been adopted to describe the Lévy stable distributions, and more generally the probability distributions governed by generalized diffusion equations of fractional order in space and/or in… (More)

- Francesco Mainardi, Gianni Pagnini
- 2007

The 1888 paper by Salvatore Pincherle (Professor of Mathematics at the University of Bologna) on generalized hypergeometric functions is revisited. We point out the pioneering contribution of the Italian mathematician towards the Mellin-Barnes integrals based on the duality principle between linear differential equations and linear difference equation with… (More)

The spectrum profile that emerges in molecular spectroscopy and atmospheric radiative transfer as the combined effect of Doppler and pressure broadenings is known as the Voigt profile function. Because of its convolution integral representation, the Voigt profile can be interpreted as the probability density function of the sum of two independent random… (More)

- Gianni Pagnini, Yangquan Chen
- 2011

Noises are usually assumed to be Gaussian so that many existing signal processing techniques can be applied with no worry. However, in many real world natural or man-made systems , noises are usually heavy-tailed. It is increasingly desirable to address the problem of finding an opportune filter function for a given input noise in order to generate a… (More)

- Antonio Mura, Gianni Pagnini
- 2008

In this paper we study a parametric class of stochastic processes to model both fast and slow anomalous diffusion. This class, called generalized grey Brownian motion (ggBm), is made up of self-similar with stationary increments processes (H-sssi) and depends on two real parameters α ∈ (0, 2) and β ∈ (0, 1]. It includes fractional Brownian motion when α ∈… (More)

- Daniel Molina-García, Tuan Minh Pham, Paolo Paradisi, Carlo Manzo, Gianni Pagnini
- Physical review. E
- 2016

We present a modeling approach for diffusion in a complex medium characterized by a random length scale. The resulting stochastic process shows subdiffusion with a behavior in qualitative agreement with single-particle tracking experiments in living cells, such as ergodicity breaking, p variation, and aging. In particular, this approach recapitulates… (More)