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Erdélyi-Kober fractional diffusion

1CRS4, Center for Advanced Studies, Research and Development in Sardinia, Polaris Bldg. 1, 09010, Pula, CA, Italy

© 2012 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Citation Information: Fractional Calculus and Applied Analysis. Volume 15, Issue 1, Pages 117–127, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: 10.2478/s13540-012-0008-1, December 2011

Publication History

Published Online:
2011-12-29

Abstract

The aim of this Short Note is to highlight that the generalized grey Brownian motion (ggBm) is an anomalous diffusion process driven by a fractional integral equation in the sense of Erdélyi-Kober, and for this reason here it is proposed to call such family of diffusive processes as Erdélyi-Kober fractional diffusion. The ggBm is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion. This class is made up of self-similar processes with stationary increments and it depends on two real parameters: 0 < α ≤ 2 and 0 < β ≤ 1. It includes the fractional Brownian motion when 0 < α ≤ 2 and β = 1, the time-fractional diffusion stochastic processes when 0 < α = β < 1, and the standard Brownian motion when α = β = 1. In the ggBm framework, the 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: Primary 26A33; Secondary 45D05, 60G22, 33E30

Keywords: anomalous diffusion; Erdélyi-Kober fractional integral and derivative; Mainardi function

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