  We derive the solution representation for a large class of nonlocal boundary
value problems for linear evolution PDEs with constant coefficients 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 different
structural properties in general.
  The nonlocal conditions we consider can be reformulated as \emph{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.
