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