Justification of the Galerkin method for hypersingular equations

The paper presents a theoretical study of hypersingular equations of the general form for problems of electromagnetic-wave diffraction on open surfaces of revolution. Justification of the Galerkin is given. The method is based on the separation of the principal term and its analytic inversion. The inverse of the principal operator is completely continuous. On the basis of this result, the equivalence of the initial equation to a Fredholm integral equation of the second kind is proven. An example of numerical solution with the use of Chebyshev polynomials of the second kind is considered.