On one method of constructing quadrature formulas for

I. V. Boykov ; A. I. Boikova

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 3, pp. 249-267

Résumé

This paper is devoted to constructing quadrature formulas for singular and hypersingular integrals evaluation. For evaluating the integrals with the weights $(1-t)^{\gamma_1}(1 + t)^{\gamma_2}$, $\gamma_1$, $\gamma_2>-1$, defined on $[-1, 1]$, we have constructed quadrature formulas uniformly converging on $[-1, 1]$ to the original integral with the weights $(1-t)^{\gamma_1}(1 + t)^{\gamma_2}$, $\gamma_1$, $\gamma_2\geqslant-1/2$, and converging to the original integral for $-1 t 1$ with the weights $(1-t)^{\gamma_1}(1 + t)^{\gamma_2}$, $\gamma_1$, $\gamma_2>-1$. In the latter case a sequence of quadrature formulas converges to evaluating integral uniformly on $[-1 + \delta, 1 -\delta]$, where $\delta>0$ is arbitrarily small. We propose a method for construction and error estimate of quadrature formulas for evaluating hypersingular integrals based on transformation of quadrature formulas for evaluation of singular integrals. We also propose a method of the error estimate for quadrature formulas for singular integrals evaluation based on the approximation theory methods. The results obtained were extended to hypersigular integrals.