Exponential asymptotics and boundary value problems: keeping both sides happy at all orders.
Exponential asymptotics and boundary value problems: keeping both sides happy at all orders.
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 ? ? 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 ? 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
exponential asymptotics, asymptotics, hyperasymptotics, transseries, boundary value problems
2771-2794
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
15 April 2010
Howls, C.J.
66d3f0f0-376c-4f7a-a206-093935e6c560
Howls, C.J.
(2010)
Exponential asymptotics and boundary value problems: keeping both sides happy at all orders.
Proceedings of the Royal Society A, 466 (2121), .
(doi:10.1098/rspa.2010.0096).
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 ? ? 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 ? 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
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Submitted date: 17 February 2010
Accepted/In Press date: 12 March 2010
e-pub ahead of print date: 8 April 2010
Published date: 15 April 2010
Keywords:
exponential asymptotics, asymptotics, hyperasymptotics, transseries, boundary value problems
Organisations:
Applied Mathematics
Identifiers
Local EPrints ID: 72516
URI: http://eprints.soton.ac.uk/id/eprint/72516
ISSN: 1364-5021
PURE UUID: 145467bb-9fcb-4aa5-adf9-ff0135742d76
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Date deposited: 17 Feb 2010
Last modified: 14 Mar 2024 02:44
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