Evolutionary equations for the theory of waveguides

A principally new approach to solving boundary value problems for Maxwell´s equations has been suggested recently by the author (see Analytical and Numerical Methods in Electromagnetic Wave Theory, Science House Co., Tokyo, 1993). Its principal advantage over the classical method of complex amplitudes of the electromagnetic field lies in the absence of restrictions upon constitutive relations which define the state of the electromagnetic field in the medium inside a cavity resonator or a waveguide. These relations may be arbitrary, i.e. both linear and nonlinear in regard to strength vectors of the electromagnetic field sought for. Another advantage of the given approach lies in opening up new possibilities for investigating transient fields directly in the time domain, i.e. without obligatory applications of Laplas´ or Fourier´s transforms. Schematically, our approach is alternative to the classical method of complex amplitudes. This scheme will be applied to the boundary value problem for a waveguide.