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In the time domain, the use of versatile finite-difference algorithms allows one to solve efficiently the problems of analysis and model synthesis, providing the following principle requirements are fulfilled. First, the analysis domain for the relevant original open boundary value problems should be restricted by exact "absorbing" conditions, which do not distort the physical processes simulated mathematically. Second, all mathematical constructions should be adequate for discretizing the problems (equations and all conditions) in rectangular coordinates with optimal and equal approximation error. The paper reviews work on algorithmization of initial boundary value problems in the theory of periodic waveguides; specifically, the case in point is the simulation of transient processes in pulse radiators. In this paper we give formulations of the initial boundary value problems and basic results for a class of antennas with gratings as dispersing elements.