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On Boundary Layer Expansions for a Singularly Perturbed Problem with Confluent Fuchsian Singularities

Laboratoire Paul Painlevé, University of Lille, 59655 Villeneuve d’Ascq CEDEX, France
Mathematics 2020, 8(2), 189; https://doi.org/10.3390/math8020189
Received: 20 December 2019 / Revised: 23 January 2020 / Accepted: 26 January 2020 / Published: 4 February 2020
(This article belongs to the Section Difference and Differential Equations)
We consider a family of nonlinear singularly perturbed PDEs whose coefficients involve a logarithmic dependence in time with confluent Fuchsian singularities that unfold an irregular singularity at the origin and rely on a single perturbation parameter. We exhibit two distinguished finite sets of holomorphic solutions, so-called outer and inner solutions, by means of a Laplace transform with special kernel and Fourier integral. We analyze the asymptotic expansions of these solutions relatively to the perturbation parameter and show that they are (at most) of Gevrey order 1 for the first set of solutions and of some Gevrey order that hinges on the unfolding of the irregular singularity for the second. View Full-Text
Keywords: asymptotic expansion; Borel–Laplace transform; Fourier transform; initial value problem; formal power series; linear integro-differential equation; partial differential equation; singular perturbation asymptotic expansion; Borel–Laplace transform; Fourier transform; initial value problem; formal power series; linear integro-differential equation; partial differential equation; singular perturbation
MDPI and ACS Style

Malek, S. On Boundary Layer Expansions for a Singularly Perturbed Problem with Confluent Fuchsian Singularities. Mathematics 2020, 8, 189.

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