We study boundary value problems for a linear evolution equation with spatial derivatives of arbitrary order, on the
domain 0 < x < L, 0 < t < T, with L and
T positive finite constants. We present a general method for identifying well-posed
problems, as well as for constructing an explicit representation of the solution of
such problems. This representation has explicit x and t dependence, and it consists
of an integral in the k-complex plane and of a discrete sum. As illustrative examples
we solve some two-point boundary value problems for the equations
iqt + qxx = 0 and
qt + qxxx = 0.