Abstract and Applied Analysis
Volume 2012 (2012), Article ID 930385, 86 pages
Research Article

On the Parametric Stokes Phenomenon for Solutions of Singularly Perturbed Linear Partial Differential Equations

UFR de Mathématiques, Université de Lille 1, 59655 Villeneuve d'Ascq Cedex, France

Received 27 March 2012; Accepted 8 July 2012

Academic Editor: Roman Dwilewicz

Copyright © 2012 Stéphane Malek. 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.


We study a family of singularly perturbed linear partial differential equations with irregular type in the complex domain. In a previous work, Malek (2012), we have given sufficient conditions under which the Borel transform of a formal solution to the above mentioned equation with respect to the perturbation parameter converges near the origin in and can be extended on a finite number of unbounded sectors with small opening and bisecting directions, say , for some integer . The proof rests on the construction of neighboring sectorial holomorphic solutions to the first mentioned equation whose differences have exponentially small bounds in the perturbation parameter (Stokes phenomenon) for which the classical Ramis-Sibuya theorem can be applied. In this paper, we introduce new conditions for the Borel transform to be analytically continued in the larger sectors , where it develops isolated singularities of logarithmic type lying on some half lattice. In the proof, we use a criterion of analytic continuation of the Borel transform described by Fruchard and Schäfke (2011) and is based on a more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above.