The unified transform method for linear initialboundary value problems: a spectral interpretation
Abstract
It is known that the unified transform method may be used to solve any wellposed initialboundary value problem for a linear constantcoefficient evolution equation on the finite interval or the halfline. In contrast, classical methods such as Fourier series and transform techniques may only be used to solve certain problems. The solution representation obtained by such a classical method is known to be an expansion in the eigenfunctions or generalised eigenfunctions of the selfadjoint ordinary differential operator associated with the spatial part of the initialboundary value problem. In this work, we emphasise that the unified transform method may be viewed as the natural extension of Fourier transform techniques for nonselfadjoint operators. Moreover, we investigate the spectral meaning of the transform pair used in the new method; we discuss the recent definition of a new class of spectral functionals and show how it permits the diagonalisation of certain nonselfadjoint spatial differential operators.
 Publication:

arXiv eprints
 Pub Date:
 August 2014
 arXiv:
 arXiv:1408.3659
 Bibcode:
 2014arXiv1408.3659S
 Keywords:

 Mathematics  Spectral Theory;
 Mathematics  Analysis of PDEs;
 35P10 (primary);
 35C15;
 35G16;
 47A70 (secondary)
 EPrint:
 3 figures