Advances in Differential Equations

Evolution PDEs and augmented eigenfunctions. Finite interval

A.S. Fokas and D.A. Smith

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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.

Article information

Adv. Differential Equations, Volume 21, Number 7/8 (2016), 735-766.

First available in Project Euclid: 3 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35P10: Completeness of eigenfunctions, eigenfunction expansions 35C15: Integral representations of solutions 35G16: Initial-boundary value problems for linear higher-order equations 47A70: (Generalized) eigenfunction expansions; rigged Hilbert spaces


Smith, D.A.; Fokas, A.S. Evolution PDEs and augmented eigenfunctions. Finite interval. Adv. Differential Equations 21 (2016), no. 7/8, 735--766.

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