2014/08/15 by David A. Smith, Smith, David A.
Mathematics · Physics and Astronomy · #35C15 #35G16 #35P10 (primary) #47A70 (secondary) #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1408.3659
openalex publication_date 2014/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is known that the unified transform method may be used to solve any well-posed initial-boundary value problem for a linear constant-coefficient evolution equation on the finite interval or the half-line. 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 self-adjoint ordinary differential operator associated with the spatial part of the initial-boundary value problem. In this work, we emphasise that the unified transform method may be viewed as the natural extension of Fourier transform techniques for non-self-adjoint 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 non-self-adjoint spatial differential operators.