Nonlocal and multipoint boundary value problems for linear evolution equations

TL;DR

This paper develops explicit solution representations for a broad class of nonlocal and multipoint boundary value problems for linear evolution PDEs, including the heat equation and third order cases, using the Fokas transform method.

Contribution

It introduces a unified approach to solve nonlocal boundary value problems for linear evolution PDEs, with explicit formulas and well-posedness criteria, especially for the less-studied third order case.

Findings

Explicit solution representations for nonlocal boundary problems.

Criteria for well-posedness in second order cases.

Application of Fokas transform method to multipoint conditions.

Abstract

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution PDEs with constant coefficients in one space variable. The prototypical such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as \emph{multipoint conditions}, and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e.\ that it admits a unique solution. For the second order case, we also give criteria that guarantee well-posedness.