Hypersingular integral operators in diffraction problems of electromagnetic waves on open surfaces

Abstract

The three-dimensional problem of diffraction of a monochromatic electromagnetic field on an open ideally conducting surface has been reduced in [1] to the solution of an integrodifferential equation for the induced density of surface currents. In [2] this equation has been represented in hypersingular form, with the divergent integral understood in the sense of Hadamard finite part. In the present paper, we examine the properties of hypersingular operators and prove boundedness of the direct operator in the Banach spaces of tangential fields with a H61der-continuous surface divergence. An algorithm for numerical solution of the hypersingular equation is proposed, based on approximation of the operator by a basis system of piecewise-linear functions. Consider the mathematical formulation of the problem of diffraction of electromagnetic waves on an ideally conducting open surface S E C 2,a with the edge C O E C 2,a, which is embedded in an infinite homogeneous isotropic medium with dielectric permittivity e, magnetic permeability /z, and conductivity a. The electromagnetic wave with harmonic time dependence and frequency c~ is described by the electric and magnetic field vectors