Unsaturated cubature formulae of hypersingular integration

Background. Hypersingular integrals are widely used in such areas as aerodynamics, theory of elasticity, electrodynamics, aerodynamics and geophysics. However, their closed form calculation is rarely possible. Therefore, using approximation methods for calculating hypersingular integrals is urgent for calculus mathematics. A lot of papers deal with this problem. I. V. Boikov and Yu. F. Zaharova have published a series of papers on constructing optimal methods for calculating hypersingular integrals. In 1975 in the USSR Academy of Sciences Proceedings (volume 221, № 1) K. I. Babenko announced the discovery of fundamentally new unsaturated numerical methods. A distinctive feature of the latter is the ability to automatically adjust to the classes of problem solving correctness. The analysis of the quadrature and cubature formulae of calculating hypersingular integrals showed that they are saturated. Therefore, it seems significant to work out unsaturated algorithms to calculate hypersingular and polyhypersingular integrals. This paper deals with this task. Materials and methods. Developing methods of calculating unsaturated hypersingular integrals is based on the constructive theory of functions and splines. Results. The optimal quadrature formulae for calculating hypersingular integrals of the same class have been derived. Unsaturated quadrature and cubature formulae for calculating one-dimensional hypersingular and polyhypersingular integrals have been derived. A comparison of the efficiency of calculating hypersingular integrals using saturated and unsaturated quadratures has been made. Conclusions. Unsaturated methods make it possible to effectively calculate hypersingular integrals while solving practical problems when a priori smoothness of integrable functions is unknown.