On a Method of Constructing Quadrature Formulas for Computing Hypersingular Integrals

Abstract

This paper is devoted to constructing quadrature formulas for evaluating singular and hypersingular integrals. For evaluating integrals with 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 weights $(1-t)^{\gamma_1} (1+t)^{\gamma_2}$, $\gamma_1,\gamma_2 \geq -1/2$ and converging to the original integral for $-1<t<1$ with 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 the integral uniformly on [$-1+\delta,1-\delta$], where $\delta>0$ is arbitrarily small. We propose a method for constructing and estimating the errors of quadrature formulas to evaluate hypersingular integrals by transforming quadrature formulas to evaluate singular integrals. We also propose a method for estimating the errors of quadrature formulas for singular integral evaluation based on approximation theory methods. The results obtained have been extended to hypersingular integrals.