Fokas diagonalization

TL;DR

This paper extends the Fokas transform method to solve a broader class of linear boundary value problems, demonstrating its diagonalization properties on semiinfinite domains, with nonlocal conditions, and mixed derivatives.

Contribution

It generalizes the Fokas diagonalization approach to new problem classes, including semiinfinite, nonlocal, and mixed derivative PDEs.

Findings

Transform pairs exhibit Fokas diagonalization in new settings

Weak diagonalization suffices for spectral transform method success

Method applies to problems with nonlocal boundary conditions

Abstract

A method for solving linear initial boundary value problems was recently reimplemented as a true spectral transform method. As part of this reformulation, the precise sense in which the spectral transforms diagonalize the underlying spatial differential operator was elucidated. That work concentrated on two point initial boundary value problems and interface problems on networks of finite intervals. In the present work, we extend these results, by means of three examples, to new classes of problems: problems on semiinfinite domains, problems with nonlocal boundary conditions, and problems in which the partial differential equation features mixed derivatives. We show that the transform pair derived via the Fokas transform method features the same Fokas diagonalization property in each of these new settings, and we argue that this weak diagonalization property is precisely that needed to ensure success of a spectral transform method.