The Over-determined Boundary Value Problem Method in the Electromagnetic Waves Propagation and Diffraction Theory

Abstract

The over-determined boundary value problems in the partial domains are proposed to be used as the auxiliary problems to investigate the wave processes in the complex structures. The necessary and su-cient conditions of solvability of the over-determined problem are the dependencies between the boundary functions. These dependencies can be obtained in terms of the Fourier transforms or Fourier coe-cients of the boundary functions. The difiraction problems for the electromagnetic waves on the conducting screens in the space and in the waveguides with metallic walls are considered as the examples. The method of partial domains is widely used in the electromagnetic wave propagation and difirac- tion theory to solve the conjugation problems and boundary value problems with mixed boundary conditions. In the case when the integral or summatorial representations of the fleld to be found are obtained in some parts of the waveguide structure it is possible to get the integral or summatorial equations equivalent to the initial problem. It is convenient to consider the over-determined boundary value problems in the partial domains as the auxiliary problems. By this we are to consider more boundary conditions on some pieces of the domains boundaries than it is necessary to choose the unique solution. The review of articles devoted to over-determined boundary value problem method for the partial difierential equations is given in the paper (1). The necessary and su-cient conditions of solvability of the over-determined problems have the form of the supplementary connections between the auxiliary boundary functions. These con- nections together with the initial boundary conditions and the conjunction conditions form the complete set of equations to determine the electromagnetic fleld. In many cases the conditions at the inflnity for the unbounded domains (the radiation conditions) can be formulated also as the auxiliary conditions for the boundary functions in the over-determined problem.