On numerical solution of hypersingular integral equations of the first kind

Sh. S. Khubezhty

Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 1, pp. 85-92

Résumé

We consider a quadrature method for the numerical solution of hypersingular integral equations on the class of functions that are unbounded at the ends of the integration interval. For a hypersingular integral with a weight function $ p (x) = 1/\sqrt{1-x^2} $, a quadrature formula of the interpolation type is constructed using the zeros of the Chebyshev orthogonal polynomial of the first kind. For a regular integral, the quadrature formula of the highest degree of accuracy is also used with the weight function $ p (x)$. After discretizing the hypersingular integral equation, the singularity parameter is given the values of the roots of the Chebyshev polynomial and, evaluating indeterminate forms when the values of the nodes coincide, a system of linear algebraic equations is obtained. But, as it turned out, the resulting system is incorrect, that is, it does not have a unique solution, there is no convergence. Due to certain additional conditions, the system turns out to be correct. This is proved on numerous test cases, in which the errors of computations are also sufficiently small. On the basis of the considered test problems, we conclude that the constructed computing scheme is convenient for implementation and effective for solving hypersingular integral equations on the class of functions of the integration interval unbound at the ends.