Singular Integral Equations and Discrete Vortices

Abstract

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.