Abstract:
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.
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