Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Double-Scale Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Linear Initial Value Problems

Version 1 : Received: 25 November 2021 / Approved: 26 November 2021 / Online: 26 November 2021 (09:56:11 CET)

How to cite: Malek, S. Double-Scale Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Linear Initial Value Problems. Preprints 2021, 2021110495 (doi: 10.20944/preprints202111.0495.v1). Malek, S. Double-Scale Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Linear Initial Value Problems. Preprints 2021, 2021110495 (doi: 10.20944/preprints202111.0495.v1).

Abstract

We examine a family of linear partial differential equations both singularly perturbed in a complex parameter and singular in complex time at the origin. These equations entail forcing terms which combine polynomial and logarithmic type functions in time and that are bounded holomorphic on horizontal strips in one complex space variable. A set of sectorial holomorphic solutions are built up by means of complete and truncated Laplace transforms w.r.t time and parameter and Fourier inverse integral in space. Asymptotic expansions of these solutions relatively to time and parameter are investigated and two distinguished Gevrey type expansions in monomial and logarithmic scales are exhibited.

Keywords

Asymptotic expansion; Borel-Laplace transform; Fourier transform; initial value problem; formal power series; singular perturbation

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