There was supplied elements of the solution theory for singular integral equations in the class of wholly integrable and non–integrable functions and the elements of the theory of the simple and double layer potential for the Helmholtz equation. These results served as a basis for reducing a wide range of boundary value problems for Laplace's equation and the Helmholtz equation and also problems in aerodynamics, electrotechnics and the theory of elasticity to boundary value singular or hypersingular integral equations. Some properties of these equations were studied.

The methods of calculations and the numerical solution (the discrete vortices method type and interpolation type) in the class of both absolutely integrable and non–integrable function were given for singular integrals and singular integral equations. These results served as a basis for the mathematical substantiation of the discrete vortices method of the numerical solution to the aerodynamic problems.

Sample calculations were given, discrete mathematical models for a wide range of problems were built: stationary and non–stationary, linear and non–linear, two–dimensional and three–dimensional problems in aerodynamics, including flows past poorly streamlined solids (i.e. solids with pointed edges, angles). Furthermore, there was also built discrete mathematical models for some two–dimensional problems in the theory of elasticity and electrostatics, which can be used as a basis for numerical experiment in these applied areas.

There was supplied calculation results for specific problems.