On a q —Analog of a Singularly Perturbed Problem of Irregular Type with Two Complex Time Variables

Alberto Lastra and Stéphane Malek
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Alberto Lastra: Departamento de Física y Matemáticas, University of Alcalá, Ap. de Correos 20, E-28871 Alcalá de Henares (Madrid), Spain
Stéphane Malek: Laboratoire Paul Painlevé, University of Lille 1, 59655 Villeneuve d’Ascq CEDEX, France

Mathematics, 2019, vol. 7, issue 10, 1-25

Abstract: The analytic solutions of a family of singularly perturbed q -difference-differential equations in the complex domain are constructed and studied from an asymptotic point of view with respect to the perturbation parameter. Two types of holomorphic solutions, the so-called inner and outer solutions, are considered. Each of them holds a particular asymptotic relation with the formal ones in terms of asymptotic expansions in the perturbation parameter. The growth rate in the asymptotics leans on the − 1 -branch of Lambert W function, which turns out to be crucial.

Keywords: asymptotic expansion; Borel–Laplace transform; Fourier transform; initial value problem; formal power series; q -difference equation; boundary layer; singular perturbation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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