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Fox \(H\) functions in fractional diffusion. (English) Zbl 1061.33012

Summary: 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.

MSC:

33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
33C20 Generalized hypergeometric series, \({}_pF_q\)
33E12 Mittag-Leffler functions and generalizations
33E20 Other functions defined by series and integrals
33E30 Other functions coming from differential, difference and integral equations
26A33 Fractional derivatives and integrals
44A15 Special integral transforms (Legendre, Hilbert, etc.)
60G18 Self-similar stochastic processes
60J60 Diffusion processes
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