In this paper, we consider new solutions and algorithms for initial boundary-value problems in the electromagnetic theory of open waveguide resonators. The approaches are based on a description of the scattering properties of such resonators in terms of the transform operators for an evolutionary basis of the non-stationary signal, that have the same significance in the course of analyses in time domain as generalized scattering matrixes in frequency domain. All suggested approaches imply using mathematically correct computational procedures at the key stages of forming the solution, in particular, the analytical regularization method-the semi-inversion method.