Approximate solution of hypersingular integral equations of the first kind with second order features on the class of functions with weight $((1+x)/(1-x))^{\pm 1/2}$

Background. Approximate methods for solving hypersingular integral equations are an actively developing area of computational mathematics. This is due to the numerous applications of hypersingular integral equations in aerodynamics, electrodynamics, physics, and the fact that analytical solutions of hypersingular integral equations are possible only in exceptional cases. In addition to direct applications in physics and technology, hypersingular integral equations of the first kind arise in the approximate solution of boundary-value problems of mathematical physics. Recently, interest in the study of analytical and numerical methods for solving hypersingular integral equations has significantly increased in connection with their active use in modeling various problems in radio engineering and radar. It turned out that one of the main methods of mathematical modeling of antennas is hypersingular integral equations. In this paper, projection methods for solving hypersingular integral equations of the first kind with second-order singularities are proposed and justified. The case when the solution has the form of $x(t)=(1-t^2)^{\pm 1/2} \phi(t)$. Materials and methods. Methods of functional analysis and approximation theory are used. Function spaces in which hypersingular operators act are introduced. To prove the solvability of the proposed computational scheme and assess the accuracy of the approximate method, the general theory of Kantorovich approximate methods is used. Results. A computational scheme is constructed for the approximate solution of hypersingular integral equations with second-order singularities on a class of solutions of the form of $x(t)=(1-t^2)^{\pm 1/2} \phi(t)$. Estimates of the speed of convergence and the error of the computational scheme are obtained. Conclusions. A computational scheme for the approximate solution of first-type hypersingular integral equations defined on a segment $[-1,1]$. The results can be used to solve problems of aerodynamics (finite-wing equation), electrodynamics (diffraction on different screens), hydrodynamics (hydrofoil theory), and to solve equations of mathematical physics by the method of boundary integral equations.