We present an algorithm for characterizing the generalized Dirichlet to Neumann map for moving initial-boundary value problems. This algorithm is derived by combining the so-called global relation, which couples the initial and boundary values of the problem, with a new method for inverting certain one-dimensional integrals. This new method is based on the spectralanalysis of an associated ordinary differantial equation and on the use of the d-bar formalism. As an illustration, the Neumann boundary value for the linearized Schrödinger equation is determined in terms of the Dirichlet boundary value and of the initial condition.
Fokas, A. S., & Pelloni, B. (2007). Generalized Dirichlet to Neumann map for moving initial-boundary value problems. Journal of Mathematical Physics, 48(1), . https://doi.org/10.1063/1.2405405