On the Parametric Stokes Phenomenon for Solutions of Singularly Perturbed Linear Partial Differential EquationsReport as inadecuate

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Abstract and Applied AnalysisVolume 2012 2012, Article ID 930385, 86 pages

Research ArticleUFR 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 inthe complex domain. In a previous work, Malek 2012, we have given sufficient conditions under which the Boreltransform of a formal solution to the above mentioned equation with respect to the perturbation parameter converges near theorigin in and can be extended on a finite number of unbounded sectors with small opening and bisectingdirections, 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 beapplied. In this paper, we introduce new conditions for the Borel transform to be analytically continuedin 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 Boreltransform described by Fruchard and Schäfke 2011 and is based on a more accurate descriptionof the Stokes phenomenon for the sectorial solutions mentioned above.

Author: Stéphane Malek

Source: https://www.hindawi.com/


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