Let q(x,t) be the solution of a given IBV problem for an evolution PDE
formulated on the half line or on a finite interval. I will discuss a
methodology which yields an explicit representation of all the
boundary values of q and of its x-derivatives in terms of the
prescribed initial and boundary conditions. I consider in detail the
case of linear problems, including linear PDE systems, and then
indicate what the results imply for the case of nonlinear equations,
most significantly for integrable equations such as the NLS and KDV
equations.