On the evaluation of certain two-dimensional singular integrals with Cauchy kernels
type: Item
instance of: scholarly article
title: On the evaluation of certain two-dimensional singular integrals with Cauchy kernels
zbMATH Open document ID: 0541.41033
DOI: 10.1007/BF01400259
published in: Numerische Mathematik
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/132986
review text: The authors consider product formulas of interpolatory type for the two- dimensional Cauchy principal value integral: \[ I(f;\eta_ 1,\eta_ 2)=\int^{1}_{-1}\int^{1}_{-1}\omega(x,y)\frac{f(x,y)}{(x-\eta_ 1)(y-\eta_ 2)}dx dy \] where: \(\eta_ 1,\eta_ 2\in(-1,1)\), \(\omega(x,y)=\omega_ 1(x).\omega_ 2(y)\) and \(\omega_ 1(x)\) and \(\omega_ 2(y)\) are two absolutely integrable weight functions. The integral is approximated by \[ F_{n,m}(f;\eta_ 1,\eta_ 2)=\sum^{n}_{i=0}\sum^{m}_{j=0}A_ i^{(1)}(\eta_ 1)A_ j^{(2)}(\eta_ 2)f(x_ i,y_ j) \] where the nodes \(\{x_ i\}\) and \(\{y_ j\}\) are the zeros of the Chebyshev polynomials of the first kind, commonly named ''classical'' abscissas, or the Clenshaw points, often called ''practical'' abscissas. Convergence results for these rules are presented.
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 3859692
zbMATH Keywords: two-dimensional singular integrals
zbMATH Keywords: Cauchy kernels
zbMATH Keywords: product formulas
zbMATH Keywords: weight functions
zbMATH Keywords: Chebyshev polynomials
author: Catterina Dagnino
author: Annamaria Palamara Orsi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3859692