Evolution PDEs and augmented eigenfunctions. Finite interval

TLDR: In this article, the authors introduce augmented eigenfunctions, a class of spectral entities for solving initial-boundary value problems of type I and type II, respectively, which can be constructed via spectral analysis.

Abstract: The so-called unified method expresses the solution of an initial-boundary value problem for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple initial-boundary value problems, 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 initialboundary value problems, which will be referred to as problems of type II, there does not exist a classical transform pair and the solution 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 introduced in the sixties by Gel’fand and his co-authors. AMS MSC2010 35P10 (primary), 35C15, 35G16, 47A70 (secondary). 1 ar X iv :1 30 3. 22 05 v2 [ m at h. SP ] 1 5 A ug 2 01 4