This monograph is divided into five parts, and opens with elements of the theory of singular integral equation solutions in the class of absolutely integrable and non-integrable functions. The second part deals with elements of potential theory for the Helmholtz equation, expecially with the reduction of Dirichlet and Neumann problems for Laplace and Helmholtz equations to singular integral equations. Part three contains methods of calculation for different one-dimensional and two-dimensional singular integrals. In this part, quadrature formulas of discrete vortex-pair types in the plane case and closed-vortex frame type in the spatial case for singular integrals are described for the first time. These quadrature formulas are applied to numerical solutions of singular integral equations of the 1st and 2nd kind with constant and variable co-efficients, in part four of the book. Finally, discrete mathematical models of some problems of aerodynamics, electrodynamics and elasticity theory are given.
This monograph should be of interest to specialists in numerical experiments in aerodynamics, elasticity theory and diffraction of waves, as well as those engaged in the theory and numerical methods in singular integral equations. The many formulations of unsolved mathematical problems contained in the book should also be of interest to postgraduate students in this field.