Some pseudodifferential operators on a segment and the theory of thin wire antennae

The problem of thin-wire antenna radiation under pulse excitation reduces to the solution of a hypersingular integral equation of the first kind on a segment, with the Dirac d-function in its right-hand side. We propose and justify the numerical method of the type of the discrete vortex pair method. We compare the numerical solution for a half-wave rectilinear cylindrical dipole with the well-known solution resulting from other considerations. We study problems of the unique existence of the solution to the corresponding characteristic hypersingular integral equation. For this purpose the equation is regarded as a pseudodifferential operator in some weight Sobolev spaces.