Evolution PDEs and augmented eigenfunctions. Half-line

TL;DR

This paper introduces augmented eigenfunctions, a new spectral concept, to construct transform pairs for solving initial-boundary value problems of evolution PDEs on the half-line, extending classical spectral methods.

Contribution

It establishes the existence and construction of non-classical transform pairs via augmented eigenfunctions for general well-posed initial-boundary value problems.

Findings

Augmented eigenfunctions form a new class of spectral functionals.

They enable the construction of transform pairs for problems lacking classical solutions.

The approach generalizes spectral methods for evolution PDEs on the half-line.

Abstract

The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the {\em unified transform} introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is {\em no} suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of {\em augmented eigenfunctions}, a novel class of spectral functionals. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.