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.
Part 1 Elements of the theory of singular integral equations: one-dimensional singular integrals; one-dimensional singular integral equations; singular integral equations with multiple Cauchy-type integrals. Part 2 Reducing of boundary problems of mathematical physics and some applied fields to the singular integral equations: boundary problems for Laplace and Helmholtz equations - plane case; boundary problems for the Laplace and the Helmholtz equations - spatial case; stationary problems of aerohydrodynamics - plan case; stationary aerohydrodynamic problems - spatial case; non-stationary aerohydrodynamic problems; determination of aerohydrodynamic characteristics; some electrostatic problems; some problems of mathematical physics; problems in elasticity theory. Part 3 Calculation of singular integral values: quadrature formulas of the method of discrete vortices for one-dimensional singular integrals; quadrature formulas of interpolation type for one-dimensional singular integrals and operators; quadrature formulas for multiple and multidimensional singular integrals; proving the Poincare-Berrand formula with the help of quadrature formulas. Part 4 Numerical solution of singular integral equations: equations of the first kind - the numerical method of discrete vortex type; equations of the first kind - interpolation methods; equations of the second kind - interpolation methods; singular integral equations with multiple Cauchy integrals. Part 5 Discrete mathematical models and calculation examples: discrete vortex systems; discrete vortex method for plance stationary problems; method of discrete vortices for spatial stationary problems; method of discrete vortices in non-stationary problems of aerodynamics; numerical method of discrete singularities in electrodynamic and elasticity theory.