On the numerical evaluation of certain 2-D singular integrals
The problem of approximating certain two-dimensional Cauchy principal value integrals is here considered, product integration formulas with multiple nodes are presented and the behaviour of the remainder is analyzed. Next a particular class of cubature rules is generated, having the peculiar property that the set of the nodes holds fixed while their multiplicities vary. Some numerical examples of application of the latter rules are also provided.
R.A. Cicenia, Numerical integration formulas involving derivatives, J. Inst. Math. Appl. 18 (1976), 79-85, https://doi.org/10.1093/imamat/18.1.79
C. Dagnino, S. Perotto and E. Santi, Product formulas based on spline approximation for the numerical evaluation of certain 2-D CPV integrals, Approx. And Optim. (Eds. D.D. Stancu, G. Coman, W.W. Breckener, P. Blaga), Transilvania Press, Cluj-Napoca (1997), 241-250.
W. Gautschi, A survey of Gauss-Christoffel quadrature formulae, G.B. Christoffel (P.L. Butzer and F. Fehèr eds.) Birkhäuser, Basel (1981), 72-147, https://doi.org/10.1007/978-3-0348-5452-8_6
A. Ghizzetti, A. Ossicini, Quadrature formulae, Birkhäuser, Basel 1970.
L. Gori, M.L. Lo Cascio, On an invariance property of the zeros of some s-orthogonal polynomials , Orthogonal polynomials and their applications (Eds. C. Brezinski, L. Gori, A. Ronveaux) J.C. Baltzer AG Scient. Publ. IMACS (1991), 277-280.
L. Gori, C.A. Micchelli, On weight functions which admit explicit Gauss-Turán quadrature formulas, Math Comp. 65 (1996), 1567-1581, https://doi.org/10.1090/s0025-5718-96-00769-7
L. Gori, E. Santi, On the evaluation of Hilbert transform by means of a particular class of Turán type quadrature rules, Numer. Algorithms 10 (1995), 27-39, https://doi.org/10.1007/bf02198294
G. Monegato, Convergence of product formulas for the numerical evaluation of certain two-dimensional Cauchy principal value integrals, Numer. Math. 43 (1984), 161-173, https://doi.org/10.1007/bf01390121
P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 213 (1979).
M.A. Sheshko, On the convergence of cubature processes for two-dimensional integrals, Dokl. Akad. Nauk. BSSR 23 (1979), 293-296.
P. Rabinowitz, Convergence results for piecewise linear quadratures for Cauchy principal value integrals, Math. Comp. 51 (1988), 741-747, https://doi.org/10.1090/s0025-5718-1988-0958639-3
D.D. Stancu, The remainder of certain linear formulas in two variables, SIAM J. Num. Anal. 1 (1964), 137-163, https://doi.org/10.1137/0701013
How to Cite
Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory
This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.