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One-dimensional stable probability density functions for rational index 0<α≤2

Hatzinikitas, A. and Pachos, J.K. (2008) One-dimensional stable probability density functions for rational index 0<α≤2. Annals of Physics, 323 (12). pp. 3000-3019. ISSN 0003-4916

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Abstract

Fox’s H-function provide a unified and elegant framework to tackle several physical phenomena. We solve the space fractional diffusion equation on the real line equipped with a delta distribution initial condition and identify the corresponding H-function by studying the small x expansion of the solution. The asymptotic expansions near zero and infinity are expressed, for rational values of the index α, in terms of a finite series of generalized hypergeometric functions. In x-space, the α=1 stable law is also derived by solving the anomalous diffusion equation with an appropriately chosen infinitesimal generator for time translations. We propose a new classification scheme of stable laws according to which a stable law is now characterized by a generating probability density function. Knowing this elementary probability density function and bearing in mind the infinitely divisible property we can reconstruct the corresponding stable law. Finally, using the asymptotic behavior of H-function in terms of hypergeometric functions we can compute closed expressions for the probability density functions depending on their parameters α β c τ. Known cases are then reproduced and new probability density functions are presented.

Item Type: Article
Copyright, Publisher and Additional Information: © 2008 Elsevier B.V. This is an author produced version of a paper published in Annals of Physics. Uploaded in accordance with the publisher's self archiving policy.
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Physics and Astronomy (Leeds)
Depositing User: Sherpa Assistant
Date Deposited: 09 Jan 2009 09:26
Last Modified: 08 Feb 2013 17:05
Published Version: http://dx.doi.org/10.1016/j.aop.2008.06.004
Status: Published
Publisher: Elsevier B.V.
Refereed: Yes
Identification Number: 10.1016/j.aop.2008.06.004
URI: http://eprints.whiterose.ac.uk/id/eprint/5299

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