Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals

Type
Article

Authors
Dujardin, G. M.

KAUST Grant Number
KUK-I1-007-43

Online Publication Date
2009-08-12

Print Publication Date
2009-11-08

Date
2009-08-12

Abstract
This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas' transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is not asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period T of the boundary data with the square of the length of the interval over. © 2009 The Royal Society.

Citation
Dujardin GM (2009) Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465: 3341–3360. Available: http://dx.doi.org/10.1098/rspa.2009.0194.

Acknowledgements
The author thanks A.S. Fokas and P.A. Markowich for their ideas and comments on this work. This publication is based on work supported by Award No. KUK-I1-007-43, made by the King Abdullah University of Science and Technology.

Publisher
The Royal Society

Journal
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

DOI
10.1098/rspa.2009.0194

Permanent link to this record