On building of quadrature and cubature formulas for computing of hyper-singular integrals

Background. There are a large number of problems, both in physics and technology, and directly in various sections of mathematics, the solution of which requires to calculate hypersingular integrals. Since direct computation of such integrals is possible only in exceptional cases, it becomes necessary to develop approximate methods. The article is devoted to the construction of approximate methods for calculating hypersynchial integrals. Particular attention is paid to the investigation of the connection between the methods for calculating singular and hypersingular integrals. Materials and methods. In this paper, we investigate the relationship between the methods for calculating singular and hyper-singular integrals. A method is proposed for estimating from above quadrature formulas for calculating hypersingular integrals and cubature formulas for calculating polyhyper-singular integrals. Results. The quadrature and cubature formulas optimal in precision (order) of the calculation of hypersingular and polyhyper-singular integrals with singularities of the second order are constructed. We consider hypersingular and polyhypersingular integrals with periodic kernels and on classes of periodic functions. Conclusions. The work proposes precision-optimal methods for calculating hypersingular and polyhyper-singular integrals that can be used for problem solving in physics, engineering and computational mathematics.