Approximate solution of some types of hypersingular integral equations

Title Approximate solution of some types of hypersingular integral equations I. Boikov1, A. Boikova11Penza State University Importance of solving hypersingular integral equations is justified by numerous applications and intense growth of the field during the last century since Hilbert and Poincare created the theory of singular integral equations. The theory is associated with numerous applications of singular and hypersingular integral equations, as well as with Riemann boundary value problem. The Riemann boundary value problem, singular, and hypersingular integral equations are broadly used as basic techniques of mathematical modeling in physics (quantum field theory, theory of short and long-range interaction, solution theory), theory of elasticity and thermoelasticity, aerodynamics and electrodynamics and many other fields. A closed-form solution of singular and hypersingular integral equations is only possible in exceptional cases. A comprehensive presentation and an extensive literature survey associated with all methods of solution of singular integral equations of the first and second kinds can be found in \cite{Goh,Mich,Boy1,Boy2}. The methods of solution of hypersingular integral equations are less elaborated \cite{Boy3}, \cite{Boy4} . In this paper,we study smoothness of solutions of hypersingular integral equations and their solvability.Also we propose an approach to approximate solving hypersingular integral equations. Using the collocation method and the method of mechanical quadrature each of these problems is approximated with systems of algebraic equations. Hypersingular integral equations, collocations, mechanical quadratures Boikov I., Boikova A. ''Approximate solution of some types of hypersingular integral equations'' [Electronic resource]. Proceedings of the XIII International scientific conference ''Differential equations and their applications in mathematical modeling''. (Saransk, July 12-16, 2017). Saransk: SVMO Publ, 2017. - pp. 446-461. Available at: http://conf.svmo.ru/files/deamm2017/papers/paper62.pdf. - Date of access: 24.10.2020.