APPROXIMATE SOLUTION OF NONLINEAR HYPERSINGULAR INTEGRAL EQUATIONS

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
Zakharova Yuliya Fridrikhovna, 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
Semov Mikhail Aleksandrovich, Postgraduate student, Penza State University (40 Krasnaya street, Penza, Russia), math@pnzgu.ru

517.392

Background. Approximate methods for solving hypersingular integral equations are an actively developing section of calculus mathematics owing to numerous applications of hypersingular integral equations in mechanics, aerodynamics, electrodynamics, geophysics. Here it is necessary to mention two important circumstances: 1) analytical solution of hypersingular integral equations is possible only in exceptional cases; 2) application range of hypersingular integral equations is constantly growing. These prove the relevance of building and substantiating numerical methods for solving hypersingular integral equations. At the present time the methods of approximate solution of nonlinear hypersingular integral equations remain undeveloped. The article is devoted to building and substantiating an approximate solution of one class of nonlinear hypersingular integral equations by the method of collocations.
Materials and methods. Substantiation of solvability and convergence of the method of collocations to the approximate solution of one class of nonlinear hypersingular integral equations, determined on closed loops, is based on application of methods of functional analysis and approximation theory.
Results. The authors suggested and substantiated the method of collocations for the approximate solution of nonlinear hypersingular integral equations, determined on closed loops. The article includes the estimates of convergence rate and extent of error.
Conclusions. The authors built a computing circuit allowing to effectively calculate applied problems of mechanics, aerodynamics, electrodynamics, geophysics.

nonlinear hypersingular integral equations, method of collocations, method of Newton-Kantorovich.

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