## 1. Introduction

[2] In this paper we describe the complicated and canonic problems and accurate solutions to them that are to create the “basis” for Time Domain (TD) system of mathematical modeling of transmission lines (resonant phenomena based units like converters, filters, hermetic matching windows, multiplexes, changers of waveguide dimensions etc) that is complementary to the developed by authors [*Shestopalov et al.*,1986a, 1986b; *Sirenko et al.*, 2000; *Yashina*, 1999; *Pochanina et al.*, 1991] and references to them) in Frequency Domain (FD). The corresponding two parts of algorithm sets (algorithms for analysis in Time and Frequency Domain) being based on rigorous mathematical models, are to create together the unique system, providing simulation of rather complicated, requiring specific accuracy, resonant electromagnetic phenomena that is only possible when carried out in parallel and interacting both in TD and FD.

[3] The partial analytical inversion of the operator by regularization of the problem is a powerful classical mathematical tool. Its gradual and well-validated application in course of developing efficient methods of the solution to homogeneous (spectral) and inhomogeneous problems of the wave diffraction theory has allowed to almost solve the problem of an adequate electromagnetic modeling for a number of canonic open resonance structures (gratings, waveguide discontinuities, open-ended screens; see, e.g. books [*Shestopalov et al.*, 1986a, 1986b] and references to them). The mathematical models of the analytical regularization methods and other rigorous analytical-numerical methods of FD have some incontrovertible advantages had been shown being worth by extensive and intensive studies of the physic nature of the resonance effects and phenomena. Unfortunately they are suitable only for a limited number of idealized scatterers having sufficiently simple geometry. The scope of problems that can be solved with these methods can be considerably enhanced by using the approaches that are based on the algorithms of the method of generalized scattering matrices. Due to these approaches, the complicated structures, whose elements are situated within a regular interaction domain of a finite length, can be efficiently analyzed, just as their separate parts. The interaction is accounted for by solving an additional, elementary, from the point of view computational complexity, problem—the canonical Fredholm set of equations with entries of the matrix operator exponentially decreasing along both the lines and columns.

[4] Method of generalized scattering matrix is also well-known in the diffraction theory, its advantages are pronounced most well only if they are thoroughly matched with the algorithms of rigorous analytical-numerical frequency domain methods, particularly, with the algorithms of the analytical regularization methods that are employed to create libraries of “elementary” units in the advanced modeling systems.

[5] Suggesting direct analogues of these powerful methods for solving problems in time domain, we hope that here in TD they will acquire same significance as their prototypes in FD. For being specific we consider as examples initial boundary-value problems for circular and coaxial waveguides and restrain the study to the simplest case of excitation of such structures by symmetrical *TE*_{0p} electromagnetic waves. Thus, the study of numerical implementation of the solution and several tests of the algorithm that are of principal importance for generic structures, are discussed in the framework of the most popular canonic problem that is a waveguide bifurcation.