APPROXIMATE SOLUTION OF HYPERSINGULAR INTEGRAL EQUATIONS ON THE NUMBER AXIS

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
Aykashev Pavel Vladimirovich, Student, Penza State University (40 Krasnaya street, Penza, Russia), math@pnzgu.ru
Semov Mikhail Aleksandrovich, Postgraduate student, Penza State University (40 Krasnaya street, Penza, Russia), math@pnzgu.ru

517.392

Background. Approximate methods of hypersingular integral equations solution are an actively developing direction in calculus mathematics, and that is first of all associated with multiple applications of hypersingular integral equations in mechanics, aerodynamics, electrodynamics, physics. At the same time, it is necessary to note the following circumstances: 1) analytical solution of hypersingular integral equations is possible only in exceptional cases; 2) the range of applications of hypersingular integral equations is constantly expanding. These aspects determine the topicality of constructing and substantiating numerical methods of hypersingular integral equations solution. At the present time the methods of approximate solution of linear and nonlinear hypersingular integral equations on the numerical axis remain undeveloped. The article is devoted to construction and substantiation of the approximate solution of one class of linear and nonlinear hypersingular integral equations on the numerical axis by the spline-collocation method with splines of zeroth order.
Materials and methods. Substantiation of solvability and convergence of the spline-collocation method to the approximate solution of one class of linear and nonlinear hypersingular integral equations, defined on the numerical axis, was based on application of methods of functional analysis and the approximation theory.
Results. The authors suggested and substantiated the spline-collocation method with splines of zeroth order for the approximate solution of linear and nonlinear hypersingular integral equations, defined on the numerical axis.
Conclusions. The researchers constructed a computing scheme allowing to effectively solve the applied problems of mechanics, aerodynamics, electrodynamics, physics.

hypersingular integral equations, spline-collocation method.

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