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Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 93502, 18 pages
http://dx.doi.org/10.1155/JAMSA/2006/93502

Explicit solutions of some fractional partial differential equations via stable subordinators

Latifa Debbi1,2,3

1Institut élie Cartan, Université Henri Poincaré – Nancy 1, B.P. 239, Vandoeuvre-lès-Nancy Cedex 54506, France
2Department of Mathematics, Faculty of Sciences, Ferhat Abbas University, El-Maabouda Sètif 19000, Algeria
3Department of Mathematics, Faculty of Sciences, University of M'sila, B.P. 166, Ichbilia, M'sila 28000, Algeria

Received 10 December 2004; Revised 5 May 2005; Accepted 10 May 2005

Copyright © 2006 Latifa Debbi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this work is to represent the solutions of one-dimensional fractional partial differential equations (FPDEs) of order (α∈ℝ+\ℕ) in both quasi-probabilistic and probabilistic ways. The canonical processes used are generalizations of stable Lévy processes. The fundamental solutions of the fractional equations are given as functionals of stable subordinators. The functions used generalize the functions given by the Airy integral of Sirovich (1971). As a consequence of this representation, an explicit form is given to the density of the 3/2-stable law and to the density of escaping island vicinity in vortex medium. Other connected FPDEs are also considered.