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- Francesco Mainardi, Gianni Pagnini
- Applied Mathematics and Computation
- 2003

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 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)

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)

- Francesco Mainardi, Gianni Pagnini, Rudolf Gorenflo
- Applied Mathematics and Computation
- 2007

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)

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)

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)

- G Pagnini, R K Saxena
- 2008

A Voigt profile function emerges in several physical investigations (e.g. atmospheric radiative transfer, astrophysical spectroscopy, plasma waves and acoustics) and it turns out to be the convolution of the Gaus-sian and the Lorentzian densities. Its relation with a number of special functions has been widely derived in literature starting from its Fourier… (More)

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… (More)

- Gianni Pagnini, Antonio Mura, Francesco Mainardi
- Philosophical transactions. Series A…
- 2013

Two-particle dispersion is investigated in the context of anomalous diffusion. Two different modelling approaches related to time subordination are considered and unified in the framework of self-similar stochastic processes. By assuming a single-particle fractional Brownian motion and that the two-particle correlation function decreases in time with a… (More)