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

TL;DR

This paper characterizes well-posed linear evolution initial-boundary value problems with arbitrary boundary conditions using Fokas' transform method, providing a concrete criterion for solution representation.

Contribution

It introduces a new, more practical criterion for well-posedness of boundary conditions in linear evolution problems, enhancing the applicability of Fokas' method.

Findings

New criterion for well-posed boundary conditions

Solution representation via series under certain conditions

Simplified test for analyticity at infinity

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.