Fox H functions in fractional diffusion

Authors:
Francesco Mainardi;Gianni Pagnini;R. K. Saxena
Affiliations:
Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy;Istituto per le Scienze dell'Atmosfera e del Clima (1SAC) del CNR, Via Gobetti 101, I-40129 Bologna, Italy;Department of Mathematics and Statistics, Jan Narain Vyas University, Jodhpur 342005, India
Venue:
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the seventh international symposium on Orthogonal polynomials, special functions and applications
Year:
2005

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Abstract

The H functions, introduced by Fox in 1961, are special functions of a very general nature, which allow one to treat several phenomena including anomalous diffusion in a unified and elegant framework. In this paper we express the fundamental solutions of the Cauchy problem for the space-time fractional diffusion equation in terms of proper Fox H functions, based on their Mellin-Barnes integral representations. We pay attention to the particular cases of space-fractional, time-fractional and neutral-fractional diffusion.