Evolution PDEs and augmented eigenfunctions. Finite interval

TL;DR

This paper introduces augmented eigenfunctions as a new spectral tool to solve complex initial-boundary value problems for evolution PDEs on finite intervals, extending classical spectral methods to problems lacking traditional transform pairs.

Contribution

It establishes the existence and construction of augmented eigenfunctions, enabling spectral analysis for problems where classical transform pairs are not applicable.

Findings

Augmented eigenfunctions exist for certain complex IBVPs.

They can be constructed via spectral analysis.

They extend spectral methods to non-classical problems.

Abstract

The so-called unified method expresses the solution of an initial-boundary value problem (IBVP) for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple IBVP, which will be referred to as problems of type~I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated IBVP, which will be referred to as problems of type~II, there does \emph{not} exist a classical transform pair and the solution \emph{cannot} be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type~II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalised eigenfunctions described in the sixties by Gel'fand and his co-authors.