Exponential asymptotics and boundary value problems: keeping both sides happy at all orders

TL;DR

This paper develops exponential asymptotic templates for boundary value problems that satisfy both boundary conditions simultaneously, revealing new transseries structures and resummation techniques for improved approximations.

Contribution

It introduces novel exponential asymptotic templates for linear and nonlinear boundary value problems, enabling simultaneous boundary satisfaction and enhanced solution approximations.

Findings

Transseries templates capture boundary conditions in singular perturbation problems.

Resummation of transseries improves approximation accuracy.

Reordering exponential asymptotics accelerates convergence.

Abstract

We introduce templates for exponential asymptotic expansions that, in contrast to matched asymptotic approaches, enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly perturbed with an asymptotic parameter epsilon \to 0+ and have a single boundary layer at one end of the interval. For linear equations, the template is a transseries that takes the form of a sliding ladder of exponential scales. For nonlinear equations, the transseries template is a two-dimensional array of exponential scales that tilts and realigns asymptotic balances as the interval is traversed. An exponential asymptotic approach also reveals how boundary value problems force the surprising presence of transseries in the linear case and negative powers of epsilon terms in the series beyond all orders in the nonlinear case. We also demonstrate how these transseries can be resummed to generate multiple-scales-type approximations that can generate uniformly better approximations to the exact solution out to larger values of the perturbation parameter. Finally we show for a specific example how a reordering of the terms in the exponential asymptotics can lead to an acceleration of the accuracy of a truncated expansion.