Exponential asymptotics and boundary value problems: keeping both sides happy at all orders - Mathematics > Classical Analysis and ODEsReport as inadecuate




Exponential asymptotics and boundary value problems: keeping both sides happy at all orders - Mathematics > Classical Analysis and ODEs - Download this document for free, or read online. Document in PDF available to download.

Abstract: We introduce templates for exponential asymptotic expansions that, incontrast to matched asymptotic approaches, enable the simultaneous satisfactionof both boundary values in classes of linear and nonlinear equations that aresingularly perturbed with an asymptotic parameter epsilon \to 0+ and have asingle boundary layer at one end of the interval. For linear equations, thetemplate is a transseries that takes the form of a sliding ladder ofexponential scales. For nonlinear equations, the transseries template is atwo-dimensional array of exponential scales that tilts and realigns asymptoticbalances as the interval is traversed. An exponential asymptotic approach alsoreveals how boundary value problems force the surprising presence oftransseries in the linear case and negative powers of epsilon terms in theseries beyond all orders in the nonlinear case. We also demonstrate how thesetransseries can be resummed to generate multiple-scales-type approximationsthat can generate uniformly better approximations to the exact solution out tolarger values of the perturbation parameter. Finally we show for a specificexample how a reordering of the terms in the exponential asymptotics can leadto an acceleration of the accuracy of a truncated expansion.



Author: C.J. Howls

Source: https://arxiv.org/







Related documents