A sincHunter quadrature rule for Cauchy principal value integrals
Author:
Bernard Bialecki
Journal:
Math. Comp. 55 (1990), 665681
MSC:
Primary 65D30
MathSciNet review:
1035926
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Abstract: A Sinc function approach is used to derive a new Hunter type quadrature rule for the evaluation of Cauchy principal value integrals of certain analytic functions. Integration over a general arc in the complex plane is considered. Special treatment is given to integrals over the interval . It is shown that the quadrature error is of order , where N is the number of nodes used, and where c is a positive constant which is independent of N. An application of the rule to the approximate solution of Cauchy singular integral equations is also discussed. Numerical examples are included to illustrate the performance of the rule.
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 [1]
 B. Bialecki and F. Stenger, SincNyström method for numerical solution of onedimensional Cauchy singular integral equation given on a smooth arc in the complex plane, Math. Comp. 51 (1988), 133165. MR 942147 (89g:65163)
 [2]
 B. Bialecki, A modified Sincquadrature rule for functions with poles near the arc of integration, BIT 29 (1989), 464476. MR 1009648 (90k:65077)
 [3]
 , SincNyström method for numerical solution of dominant system of Cauchy singular integral equations given on a piecewise smooth contour, SIAM J. Numer. Anal. 26 (1989), 11941211. MR 1014882 (90m:65225)
 [4]
 D. Elliott and D. F. Paget, Gauss type quadrature rules for Cauchy principal value integrals, Math. Comp. 33 (1979), 301309. MR 514825 (81h:65023)
 [5]
 D. Elliott, The classical collocation method for singular integral equations, SIAM J. Numer. Anal. 19 (1982), 816832. MR 664887 (83f:65208)
 [6]
 D. Elliott and F. Stenger, Sinc method of solution of singular integral equations, IMACS Symposium on Numerical Solution of Singular Integral Equations, IMACS, 1984, pp. 2735.
 [7]
 W. Gautschi and J. Wimp, Computing the Hilbert transform of a Jacobi weight function, BIT 27 (1987), 203215. MR 894123 (88i:65154)
 [8]
 A. Gerasoulis, Piecewisepolynomial quadratures for Cauchy singular integrals, SIAM J. Numer. Anal. 23 (1986), 891902. MR 849289 (87m:65039)
 [9]
 M. A. Golberg, The numerical solution of Cauchy singular integral equations with constant coefficients, J. Integral Equations (Suppl.) 9 (1985), 127151. MR 792421 (87g:65163)
 [10]
 D. B. Hunter, Some Gausstype formulae for the evaluation of Cauchy principal values of integrals, Numer. Math. 19 (1972), 419424. MR 0319355 (47:7899)
 [11]
 N. I. Muskhelishvili, Singular integral equations, Noordhoff, Groningen, 1958. MR 0355494 (50:7968)
 [12]
 F. Stenger, Approximations via Whittaker's cardinal function, J. Approx. Theory 17 (1976), 222240. MR 0481786 (58:1885)
 [13]
 , Numerical methods based on Whittaker cardinal, or Sinc functions, SIAM Rev. 23 (1981), 165224. MR 618638 (83g:65027)
 [14]
 E. Venturino, On solving integral equations via a hyperbolic tangent quadrature rule, Math. Comp. 47 (1986), 159167. MR 842128 (87k:65158)
 [15]
 S. Welstead, Orthogonal polynomials applied to the solution of singular integral equations, Ph.D. Thesis, Purdue Univ., West Lafayette, IN, 1982.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199010359263
PII:
S 00255718(1990)10359263
Keywords:
Gauss quadratures,
Cauchy singular integral equations
Article copyright:
© Copyright 1990
American Mathematical Society
