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.