On Gevrey asymptotics for linear singularly perturbed equations with linear fractional transforms
Identifiers
Permanent link (URI): http://hdl.handle.net/10017/49350DOI: 10.1007/s13398-021-01064-w
ISSN: 1578-7303
Publisher
Springer Nature
Date
2021-05-19Bibliographic citation
Chen, G, Lastra, A. & Malek, S. 2021, "On Gevrey asymptotics for linear singularly perturbed equations with linear fractional transforms", RACSAM, vol. 115, art. 121.
Keywords
Asymptotic expansion
Lambert W function
Borel-Laplace transform
Fourier transform
Initial value problem
Formal power series
Singular perturbation
Document type
info:eu-repo/semantics/article
Version
info:eu-repo/semantics/acceptedVersion
Publisher's version
https://doi.org/10.1007/s13398-021-01064-wRights
Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
Access rights
info:eu-repo/semantics/openAccess
Abstract
A family of linear singularly perturbed Cauchy problems is studied. The equations defining the problem combine both partial differential operators together with the action of linear fractional transforms.
The exotic geometry of the problem in the Borel plane, involving both sectorial regions and strip-like
sets, gives rise to asymptotic results relating the analytic solution and the formal one through Gevrey
asymptotic expansions. The main results lean on the appearance of domains in the complex plane which
remain intimately related to Lambert W function, which turns out to be crucial in the construction of
the analytic solutions.
On the way, an accurate description of the deformation of the integration paths defining the analytic
solutions and the knowledge of Lambert W function are needed in order to provide the asymptotic
behavior of the solution near the origin, regarding the perturbation parameter. Such deformation varies
depending on the analytic solution considered, which lies in two families with different geometric features.
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