Boundary layer expansions for initial value problems with two complex time variables

A Lastra,S Malek

doi:10.1186/s13662-020-2496-3

Abstract

We study a family of partial differential equations in the complex domain, under the action of a complex perturbation parameter ϵ. We construct inner and outer solutions of the problem and relate them to asymptotic representations via Gevrey asymptotic expansions with respect to ϵ in adequate domains. The asymptotic representation leans on the cohomological approach determined by the Ramis–Sibuya theorem.

Highlights

1 Introduction The main aim in this work is to describe the analytic solutions and asymptotic behavior of the solutions of a family of initial value problems in the complex domain. Such a family consists of partial differential equations in two complex time variables of the form
Q(X) ∈ C[X] and P(T1, T2, Z, z, ) stands for a polynomial in (T1, T2, Z) with holomorphic coefficients w.r.t. (z, ) on Hβ × D(0, 0), where Hβ stands for the horizontal strip in the complex plane
The symbol acts as a small complex perturbation parameter in the equation

Summary

Introduction

The main aim in this work is to describe the analytic solutions and asymptotic behavior of the solutions of a family of initial value problems in the complex domain. We detail the elements involved in the main problem under study and provide different approaches which might be followed in order to search analytic and asymptotically related formal solutions.

Results

Conclusion