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Evolution PDEs and augmented eigenfunctions. Finite interval. (English) Zbl 1375.35283

Summary: The so-called unified or Fokas 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 IBVPs, 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 IBVPs, 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 generalized eigenfunctions described in the sixties by Gel’fand and his co-authors.

MSC:

35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35C15 Integral representations of solutions to PDEs
35G16 Initial-boundary value problems for linear higher-order PDEs
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
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