Strongly regular multi-level solutions of singularly perturbed linear partial differential equations

TL;DR

This paper investigates the asymptotic behavior of solutions to singularly perturbed PDEs in the complex domain, introducing a multi-level Ramis-Sibuya theorem to handle different asymptotic levels associated with strongly regular sequences.

Contribution

It presents a novel multi-level Ramis-Sibuya theorem and applies it to analyze the asymptotic structure of solutions to singularly perturbed PDEs.

Findings

Asymptotic solutions are represented by formal power series.

Different asymptotic levels are characterized by strongly regular sequences.

The multi-level Ramis-Sibuya theorem provides a new analytical framework.

Abstract

We study the asymptotic behavior of the solutions related to a family of singularly perturbed partial differential equations in the complex domain. The analytic solutions are asymptotically represented by a formal power series in the perturbation parameter. The geometry of the problem and the nature of the elements involved in it give rise to different asymptotic levels related to the so-called strongly regular sequences. The result leans on a novel version of a multi-level Ramis-Sibuya theorem.