Optimal with respect to accuracy methods for evaluating hypersingular integrals

I. V. Boykov ; A. I. Boikova

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 23 (2021) no. 4, pp. 360-378

Résumé

In this paper we constructed optimal with respect to order quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions $\Omega_{r,\gamma}^{u}(\Omega,M),$ $\bar \Omega_{r,\gamma}^{u}(\Omega,M)$, $\Omega=[-1,1]^l,$ $l=1,2,\ldots,M=Const,$ and $\gamma$ is a real positive number. The functions that belong to classes $\Omega_{r,\gamma}^{u}(\Omega,M)$ and $\bar \Omega_{r,\gamma}^{u}(\Omega,M)$ have bounded derivatives up to the $r$th order in domain $\Omega$ and derivatives up to the $s$th order $(s=r+\lceil \gamma \rceil)$ in domain $\Omega \backslash \Gamma,$ $\Gamma = \partial \Omega.$ Moduli of derivatives of the $v$th order $(r v \le s)$ are power functions of $d(x,\Gamma)^{-1}(1+|\ln d(x,\Gamma)|),$ where $d(x,\Gamma)$ is a distance between point $x$ and $\Gamma.$ The interest in these classes of functions is due to the fact that solutions of singular and hypersingular integral equations are their members. Moreover various physical fields, in particular gravitational and electromagnetic fields belong to these classes as well. We give definitions of optimal with respect to accuracy methods for solving hypersingular integrals. We constructed optimal with respect to order of accuracy quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions $\Omega_{r,\gamma}^{u}(\Omega,M)$ and $\bar \Omega_{r,\gamma}^{u}(\Omega,M)$.

Détail

Keywords: hypersingular integrals, optimal methods.
Mots-clés : quadrature formulas