On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives
Author:
N. I. Ioakimidis
Journal:
Math. Comp. 44 (1985), 191198
MSC:
Primary 65D32; Secondary 65R20
MathSciNet review:
771040
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Abstract: The convergence of the aforementioned quadrature rules for integrands possessing Holdercontinuous derivatives of an appropriate order is proved to be uniform and not only pointwise. The rate of convergence is also established and an application to the numerical solution of singular integral equations is made.
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 [1]
 P. J. Davis & P. Rabinowitz, Methods of Numerical Integration, 1st ed., Academic Press, New York, 1975. MR 0448814 (56:7119)
 [2]
 D. Elliott, "On the convergence of Hunter's quadrature rule for Cauchy principal value integrals," BIT, v. 19, 1979, pp. 457462. MR 559954 (81m:65031)
 [3]
 D. Elliott & D. F. Paget, "On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals," Numer. Math., v. 23, 1975, pp. 311319. MR 0380215 (52:1115)
 [4]
 D. Elliott & D. F. Paget, "On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals: an addendum," Numer. Math., v. 25, 1976, pp. 287289. MR 0411131 (53:14870)
 [5]
 D. Elliott & D. F. Paget, "Gauss type quadrature rules for Cauchy principal value integrals," Math. Comp., v. 33, 1979, pp. 301309. MR 514825 (81h:65023)
 [6]
 D. B. Hunter, "Some Gausstype formulae for the evaluation of Cauchy principal values of integrals," Numer. Math., v. 19, 1972, pp. 419424. MR 0319355 (47:7899)
 [7]
 N. I. Ioakimidis, General Methods for the Solution of Crack Problems in the Theory of Plane Elasticity, Doctoral Thesis at the National Technical University of Athens, Athens, 1976. [Available from University Microfilms, Ann Arbor, Michigan; order no. 7621,056.] (Greek) MR 0521392 (58:25233)
 [8]
 N. I. Ioakimidis, "On the numerical evaluation of derivatives of Cauchy principal value integrals," Computing, v. 27, 1981, pp. 8188. MR 623178 (83g:65023)
 [9]
 N. I. Ioakimidis, "Further convergence results for two quadrature rules for Cauchy type principal value integrals," Apl. Mat., v. 27, 1982, pp. 457466. MR 678115 (84f:65017)
 [10]
 N. I. Ioakimidis, "On the numerical evaluation of a class of finitepart integrals," Z. Angew. Math. Mech., v. 63, 1983, pp. 572574. MR 729122 (85j:65011)
 [11]
 N. I. Ioakimidis & P. S. Theocaris, "On the numerical evaluation of Cauchy principal value integrals," Rev. Roumaine Sci. Tech. Sér. Méc. Appl., v. 22, 1977, pp. 803818. MR 0483321 (58:3334)
 [12]
 N. I. Ioakimidis & P. S. Theocaris, "On the numerical solution of singular integrodifferential equations," Quart. Appl. Math., v. 37, 1979, pp. 325331. MR 548991 (81a:65124)
 [13]
 A. I. Kalandiya, Mathematical Methods of TwoDimensional Elasticity, 1st English ed., p. 108, "Mir", Moscow, 1975. MR 0400846 (53:4676)
 [14]
 A. A. Korneĭchuk, "Quadrature formulae for singular integrals," Zh. Vychisl. Mat. i Mat. Fiz., v. 4, no. 4, suppl., 1964, pp. 6474. (Russian) MR 0175299 (30:5484)
 [15]
 H. R. Kutt, On the Numerical Evaluation of FinitePart Integrals Involving an Algebraic Singularity, CSIR Special Report WISK 179, National Research Institute for Mathematical Sciences, P.O. Box 395, Pretoria 0001, South Africa. [Available also from NTIS (National Technical Information Services), U.S.A.]
 [16]
 G. Monegato, "The numerical evaluation of onedimensional Cauchy principal value integrals," Computing, v. 29, 1982, pp. 337354. MR 684742 (84c:65044)
 [17]
 N. I. Muskhelishvili, Singular Integral Equations, 2nd English ed., p. 228, WoltersNoordhoff, Groningen, 1958. MR 0355494 (50:7968)
 [18]
 D. F. Paget, "The numerical evaluation of Hadamard finitepart integrals," Numer. Math., v. 36, 1981, pp. 447453. MR 614859 (83d:65062)
 [19]
 D. F. Paget & D. Elliott, "An algorithm for the numerical evaluation of certain Cauchy principal value integrals," Numer. Math., v. 19, 1972, pp. 373385. MR 0366004 (51:2256)
 [20]
 D. G. Sanikidze, "On a uniform estimation of approximation of singular integrals with Chebyshev's weighted function by sums of interpolating type," Soobshch. Akad. Nauk Gruzin. SSR, v. 75, no. 1, 1974, pp. 5355. (Russian) MR 0358242 (50:10707)
 [21]
 M. A. Sheshko, "On the convergence of quadrature processes for a singular integral," Soviet Math. (Iz. VUZ), v. 20, no. 12, 1976, pp. 8694. (English)
 [22]
 G. J. Tsamasphyros & P. S. Theocaris, "On the convergence of a Gauss quadrature rule for evaluation of Cauchy type singular integrals," BIT, v. 17, 1977, pp. 458464. MR 0468120 (57:7959)
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DOI:
http://dx.doi.org/10.1090/S00255718198507710408
PII:
S 00255718(1985)07710408
Article copyright:
© Copyright 1985
American Mathematical Society
