On the numerical solution of a hypersingular integral equation on the circle with a singular and a weakly singular term

Abstract

INTRODUCTION The notion of spaces of fractional ratios of periodic functions was introduced and discrete operators in these spaces were studied in [1, 2]. This theory was used in [3] for the investigation of the scheme based on the method of discrete vortical pairs for the solution of the hypersingular and singular integral equations on a circle to which the Neumann problem for the Laplace equation with a closed contour can be reduced by means of a double-layer potential. In the present paper, we use which theory for the analysis of a numerical scheme for a hypersingular integral equation on the circle with a singular and a weakly singular term. A similar problem arises in antenna theory [4]. We give the rate of convergence of the approximate solution to the exact solution. Thus, in the present paper, we consider the equation