  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.
