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), 191-198

MSC:
Primary 65D32; Secondary 65R20

MathSciNet review:
771040

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The convergence of the aforementioned quadrature rules for integrands possessing Holder-continuous 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.

**[1]**Philip J. Davis and Philip Rabinowitz,*Methods of numerical integration*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers] New York-London, 1975. Computer Science and Applied Mathematics. MR**0448814****[2]**David Elliott,*On the convergence of Hunter’s quadrature rule for Cauchy principal value integrals*, BIT**19**(1979), no. 4, 457–462. MR**559954**, 10.1007/BF01931261**[3]**David Elliott and D. F. Paget,*On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals*, Numer. Math.**23**(1975), 311–319. MR**0380215****[4]**David Elliott and D. F. Paget,*“On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals” (Numer. Math. 23 (1975), 311–319): an addendum*, Numer. Math.**25**(1975/76), no. 3, 287–289. MR**0411131****[5]**David Elliott and D. F. Paget,*Gauss type quadrature rules for Cauchy principal value integrals*, Math. Comp.**33**(1979), no. 145, 301–309. MR**514825**, 10.1090/S0025-5718-1979-0514825-2**[6]**D. B. Hunter,*Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals*, Numer. Math.**19**(1972), 419–424. MR**0319355****[7]**Nikolaos I. Ioakimidis,*General methods for the solution of crack problems in the theory of plane elasticity*, Leousēs Mastrogiannēs, Athens, 1976. With an English summary. MR**0521392****[8]**N. I. Ioakimidis,*On the numerical evaluation of derivatives of Cauchy principal value integrals*, Computing**27**(1981), no. 1, 81–88 (English, with German summary). MR**623178**, 10.1007/BF02243440**[9]**N. I. Ioakimidis,*Further convergence results for two quadrature rules for Cauchy type principal value integrals*, Apl. Mat.**27**(1982), no. 6, 457–466 (English, with Czech summary). With a loose Russian summary. MR**678115****[10]**N. I. Ioakimidis,*On the numerical evaluation of a class of finite-part integrals*, Z. Angew. Math. Mech.**63**(1983), no. 11, 572–574. MR**729122**, 10.1002/zamm.19830631108**[11]**N. I. Ioakimidis and P. S. Theocaris,*On the numerical evaluation of Cauchy principal value integrals*, Rev. Roumaine Sci. Tech. Sér. Méc. Appl.**22**(1977), no. 6, 803–818. MR**0483321****[12]**N. I. Ioakimidis and P. S. Theocaris,*On the numerical solution of singular integro-differential equations*, Quart. Appl. Math.**37**(1979/80), no. 3, 325–331. MR**548991****[13]**A. I. Kalandiya,*Mathematical methods of two-dimensional elasticity*, Mir Publishers, Moscow, 1975. Translated from the Russian by M. Konyaeva [M. Konjaeva]. MR**0400846****[14]**A. A. Korneĭčuk,*Quadrature formulae for singular integrals*, Ž. Vyčisl. Mat. i Mat. Fiz.**4**(1964), no. 4, suppl., 64–74 (Russian). MR**0175299****[15]**H. R. Kutt,*On the Numerical Evaluation of Finite-Part 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 one-dimensional Cauchy principal value integrals*, Computing**29**(1982), no. 4, 337–354 (English, with German summary). MR**684742**, 10.1007/BF02246760**[17]**N. I. Muskhelishvili,*Singular integral equations*, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR**0355494****[18]**D. F. Paget,*The numerical evaluation of Hadamard finite-part integrals*, Numer. Math.**36**(1980/81), no. 4, 447–453. MR**614859**, 10.1007/BF01395957**[19]**D. F. Paget and David Elliott,*An algorithm for the numerical evaluation of certain Cauchy principal value integrals*, Numer. Math.**19**(1972), 373–385. MR**0366004****[20]**D. G. Sanikidze,*The uniform estimation of the approximation of singular integrals with a Čebyšev weight function by sums of interpolation type*, Sakharth. SSR Mecn. Akad. Moambe**75**(1974), 53–55 (Russian, with Georgian and English summaries). MR**0358242****[21]**M. A. Sheshko, "On the convergence of quadrature processes for a singular integral,"*Soviet Math.*(*Iz. VUZ*), v. 20, no. 12, 1976, pp. 86-94. (English)**[22]**G. J. Tsamasphyros and P. S. Theocaris,*On the convergence of a Gauss quadrature rule for evaluation of Cauchy type singular integrals*, Nordisk Tidskr. Informationsbehandling (BIT)**17**(1977), no. 4, 458–464. MR**0468120**

Retrieve articles in *Mathematics of Computation*
with MSC:
65D32,
65R20

Retrieve articles in all journals with MSC: 65D32, 65R20

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1985-0771040-8

Article copyright:
© Copyright 1985
American Mathematical Society