On a method of numerical solution of singular integrodifferential equations (Q1082789)

Many problems of mathematical physics, in particular the diffraction problem for electromagnetic waves on ideally conducting screens, reduce to the solution of singular integrodifferential equations of the form \[ \frac{d}{dx}\int^{b}_{-b}\frac{\phi (t)}{t-x}dt+\int^{b}_{- b}K(x,t)\phi (t)dt=f(x). \] \textit{A. G. Davydov}, the first author, and \textit{Yu. V. Pimerov} [Dokl. Akad. Nauk SSSR 261, No.2, 338-341 (1981)] presented and realized an algorithm for the numerical solution of equations of this type, based on the method of piecewise constant approximation and collocation. In the present paper unique solvability is proved for the obtained system of linear algebraic equations. A uniform estimate of the speed of convergence of the approximate solution to the exact solution is obtained.