Nonlocal and multipoint boundary value problems for linear evolution equations

We derive the solution representation for a large class of nonlocal boundary
value problems for linear evolution PDEs with constant coecients
in one space variable. The prototypical such PDE is the heat equation,
for which problems of this form model physical phenomena in chemistry
and for which we formulate and prove a full result. We also consider the
third order case, which is much less studied and has been shown by the
authors to have very dierent structural properties in general.
The nonlocal conditions we consider can be reformulated as multipoint
conditions, and then an explicit representation for the solution of the
problem is obtained by an application of the Fokas transform method.
The analysis is carried out under the assumption that the problem being
solved is well posed, i.e. that it admits a unique solution. For the second
order case, we also give criteria that guarantee well-posedness.