Pproximate methods of solving hypersingular integral equations of first kind with second-order peculiarities on classes of functions with weights $(1-t^2)^{-1/2}$

Background. Approximate methods of solving hypersingular integral equations appear to be an actively developing area of calculus mathematics. It is associated, first of all, with multiple applications of hypersingular integral equations in mechanics, aerodynamics, electrodynamics, geophysics. At the same time it is necessary to point out two circumstances: analytical solution of hypersingular integral equations is possible only in exceptional cases; the range of applications of hypersingular integral equations is constantly expanding. In this regard, the development and substantiation of numerical methods of solving hypersingular integral equations are topical. At the present time, methods of approximate solution of hypersingular integral equations of first kind on classes of functions with weights $(1-t^2)^{-1/2}$ are remaining undeveloped. The article is devoted to construction and substantiation of approximate solution of hypersingular integral equations of first kind, determined on the segment [-1,1] by the method of mechanical quadrature. The authors studied methods of approximate solution of hypersingular integral equations of first kind on lcasses of functions with weights $(1-t^2)^{-1/2}$. Materials and methods. The substantiation of solvability and convergence of the method of mechanical quadratures to the approximate solution of hypersingular equations of first kind, determined on the segment [-1,1], is based on application of methods of functional analysis and the theory of approximations. Results. The paper suggests and substantiates the method of mechanical quadratures for approaximate solution of hypersingular integral equations of first kind on classes of functions with weights $(1-t^2)^{-1/2}$. Estimates of the rate of convergence and the extent of error are also adduced. Conclusions. The authors have constructed computing schemes allowing to efficiently solve applied problems of mechanics, aerodynamics, electrodynamics, geophysics.