Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Well-posed two-point initial-boundary value problems with arbitrary boundary conditions

DAVID A. SMITHa1

a1 Department of Mathematics, University of Reading RG6 6AX. e-mail: d.a.smith@reading.ac.uk

Abstract

We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series.

The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.

(Received May 05 2011)

(Revised September 27 2011)

(Online publication December 12 2011)