Method of Discrete Vortices presents a mathematical substantiation and in-depth description of numerical methods for solving singular integral equations with one-dimensional and multiple Cauchy integrals. The book also presents the fundamentals of the theory of singular equations and numerical methods as applied to solving problems in such branches of mechanics as aerodynamics, elasticity, and electrodynamics.
An Introduction to Singular Integral Equations in Aerodynamics. Quadrature Formulas for Singular Integrals: Quadrature Formulas of the Method of Discrete Vortices for One-Dimensional Singular Integrals. Interpolation Quadrature Formulas for One-Dimensional Singular Integrals. Quadrature Formulas for Multiple and Multidimensional Singular Integrals. Poincaré-Bertrand Formula. Numerical Solution of Singular Integral Equations: Equation of the First Kind on a Segment and/or a System of Nonintersecting Segments. Equations of the First Kind on a Circle Containing Hilbert's Kernel. Singular Integral Equations of the Second Kind. Singular Integral Equations with Multiple Cauchy Integrals. Application of the Method of Discrete Vortices to Aerodynamics: Verification of the Method: Formulation of Aerodynamic Problems and Discrete Vortex Systems. Two-Dimensional Problems for Airfoils. Three-Dimensional Problems. Unsteady Linear and Nonlinear Problems. Aerodynamic Problems for Blunt Bodies. Some Questions of Regularization in the Method of Discrete Vortices and Numerical Solution of Singular Integral Equations. Some Problems of the Theory of Elasticity, Electrodynamics, and Mathematical Physics: Singular Integral Equations of the Theory of Elasticity. Numerical Method of Discrete Singularities in Boundary Value Problems of Mathematical Physics. Reduction of Some Boundary Value Problems of Mathematical Physics to Singular Integral Equations.