ON ONE APPROXIMATE METHOD OF SOLVING LINEAR HYPERSINGULAR INTEGRAL
EQUATIONS ON OPEN INTEGRATION CONTOURS

Boykov Il'ya Vladimirovich, Doctor of physical and mathematical sciences, professor, head of sub-department of higher and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), boikov@pnzgu.ru
Boykova Alla Il'inichna, Candidate of physical and mathematical sciences, associate professor, sub-department of higher and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), math@pnzgu.ru

517.392

10.21685/2072-3040-2016-2-3

Background. Approximate methods of solving hypersingular integral equations are an actively developing area of calculus mathematics. This fact relates to multiple applications of hypersingular integral equations in aerodynamics, electrodynamics, physics, and also to the fact that analytical solutions of hypersingular integral equations are possible only in exceptional cases. Apart from direct applications in physics and engineering, hypersingular integral equations occur in approximate solution of boundary problems of mathematical physics. Recently, there have been published several works, devoted to approximate methods of solving hypersingular integral equations of the first kind on open integration contours. The interest to such equations is associated with their direct applications in aerodynamics and electrodynamics. Boundary conditions are used in all those works when building computing schemes. The present work suggests general methods of solving hypersingular integral equations of the first and second kinds on open integration contours. The authors have obtained values of convergency rapidity and error.
Materials and methods. In the study the authors used methods of functional analysis and approximation theory. The researchers considered linear onedimensionalhypersingular integral equations on open integration contours and built projection computing schemes, which were substantiated on the basis of the general theory of approximate methods of L. V. Kantorovich.
Results. The authors built computing schemes of collocation and mechanical quadrature methods of solving hypersingular integral equations on open integration contours and obtained values of computing schemes’ convergence rapidity and error.
Conclusions. The researchers built and substantiated computing schemes of approximate solution of hypersingular integral equations, determined on the segement [–1,1]. The obtained results may be used in solving problems of aerodynamics, electrodynamics, in solving equations of mathematical physics by the methods of boundary integral equations.

hypersingular integral equations, computing scheme, boundary conditions, open integration contour.

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