One multidimensional weakly singular integral equation

Abstract

where D is the unit circle with the center at the origin of coordinates, x = (x1, x2), y = (y1, y2) are its points, ρ(x, y) = |x− y| is the Euclidean distance between these points, a(x) ∈ C(D), h(x, y) ∈ C(D ×D), f(x) ∈ L2(D) are given functions, and φ(x) ∈ L2(D) is the desired one. As a rule, one cannot solve this equation exactly and therefore has to apply approximative solution methods. In papers [3]–[6] one proposes and substantiates approximative solution methods for multidimensional singular integral equations with a parallelepiped singularity (see also [7] and references therein). However, the possibility of the numerical solution of singular integral equations with a spherical singularity, including equation (1), is not established yet. In [2] one obtains only some results, concerning the approximate solution of one particular case of singular integral equation (1). In paper [8] one investigates a multidimensional equation of form (1) for an arbitrary open bounded domain with a piecewise-smooth boundary. In the mentioned paper, under rather rigid conditions imposed on the known functions one theoretically justifies the methods of collocations and mechanical cubages constructed on the base of a piecewise constant approximation, assuming the existence and the uniqueness of a solution to the equation. Though one has studied multidimensional equations of the form (1), many methods for their numerical solution are not justified yet. Therefore, the approximate solution of equation (1) with the corresponding function-theoretic substantiation still remains an actual problem. In this paper we establish sufficient conditions for the positive definiteness of the operator A in the space L2. On the base of these conditions we deduce the theorems on the existence and the uniqueness of a solution to singular integral equation (1). In addition, we propose numerical schemes and theoretic substantiation of approximative solution methods for singular integral equation (1).