Approximate solution of hypersingular integral equations of first kind

Background. Approximate methods of hypersingular integral equations solution are an actively developing field of calculus mathematics. It is associated with multiple applications of hypersingular integral equations in aerodynamics, electrodynamics, physics and with a circumstance, when analytical solutions of hypersingular integral equations are possible only in exceptional cases. Besides direct applications in physics and engineering, hypersingular integral equations of first kind occur at approximate solution of boundary problems of mathematical physics. At the present time there are no substantiations of numerical methods of hypersingular integral equations solution at various features on ends of a segment, on which an equation is set up. The present work suggests and substantiates a method of mechanical quadratures of solution of hypersingular integral equations of first kind at all possible features on ends of the segment [-1,1]. Materials and methods. The research included the methods of functional analysis and approximation theory. The authors considered three known classes of solution of hypersingular integral equations of first kind: a class of solutions that turn to zero on both ends of the segment [-1,1]; a class of solutions that turn to infinity on both ends of the segment [-1,1]; a class of solutions that turn to zero on one end of the segment [-1,1] and to infinity on another. Approximate solutions of equations are sought in the forms of Chebyshev's polynomials of first and second kinds, and the method of mechanical quadratures is substantiated on the basis of the general theory of approximate methods. Results. The authors have built three computing circuits for solving hypersingular integral equations of first kind. Each computing circuit has been designed for solving hypersingular integral equations with preset features of solutions on ends of the segment [-1,1]. The work describes estimates of the methods' rapidity of converegence and errors. Conclusions. The authors have built and substantiated computing circuits for approximate solution of hypersingular integral equations of first kind, determined on the segment [-1,1]. The obtained results may be used in solving problems of aerodynamics (finite wing equations), electrodynamics (diffraction on various screens), hydrodynamics (hydrofoil theory), in solving mathematical physics equations by the method of boundary integral equations.