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

G. M. Dujardin

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)3341-3360
Number of pages20
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume465
Issue number2111
DOIs
StatePublished - Aug 12 2009
Externally publishedYes

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