Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 93502, 18 pages
doi:10.1155/JAMSA/2006/93502
Explicit solutions of some fractional partial
differential equations via stable subordinators
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.