A sinc-Hunter quadrature rule for Cauchy principal value integrals

Author:
Bernard Bialecki

Journal:
Math. Comp. **55** (1990), 665-681

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1990-1035926-3

MathSciNet review:
1035926

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Bernard Bialecki and Frank Stenger,*Sinc-Nyström method for numerical solution of one-dimensional Cauchy singular integral equation given on a smooth arc in the complex plane*, Math. Comp.**51**(1988), no. 183, 133–165. MR**942147**, https://doi.org/10.1090/S0025-5718-1988-0942147-X**[2]**Bernard Bialecki,*A modified sinc quadrature rule for functions with poles near the arc of integration*, BIT**29**(1989), no. 3, 464–476. MR**1009648**, https://doi.org/10.1007/BF02219232**[3]**Bernard Bialecki,*Sinc-Nyström method for numerical solution of a dominant system of Cauchy singular integral equations given on a piecewise smooth contour*, SIAM J. Numer. Anal.**26**(1989), no. 5, 1194–1211. MR**1014882**, https://doi.org/10.1137/0726067**[4]**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**, https://doi.org/10.1090/S0025-5718-1979-0514825-2**[5]**David Elliott,*The classical collocation method for singular integral equations*, SIAM J. Numer. Anal.**19**(1982), no. 4, 816–832. MR**664887**, https://doi.org/10.1137/0719057**[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. 27-35.**[7]**Walter Gautschi and Jet Wimp,*Computing the Hilbert transform of a Jacobi weight function*, BIT**27**(1987), no. 2, 203–215. MR**894123**, https://doi.org/10.1007/BF01934185**[8]**Apostolos Gerasoulis,*Piecewise-polynomial quadratures for Cauchy singular integrals*, SIAM J. Numer. Anal.**23**(1986), no. 4, 891–902. MR**849289**, https://doi.org/10.1137/0723057**[9]**Michael A. Golberg,*The numerical solution of Cauchy singular integral equations with constant coefficients*, J. Integral Equations**9**(1985), no. 1, suppl., 127–151. MR**792421****[10]**D. B. Hunter,*Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals*, Numer. Math.**19**(1972), 419–424. MR**0319355**, https://doi.org/10.1007/BF01404924**[11]**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****[12]**Frank Stenger,*Approximations via Whittaker’s cardinal function*, J. Approximation Theory**17**(1976), no. 3, 222–240. MR**0481786****[13]**Frank Stenger,*Numerical methods based on Whittaker cardinal, or sinc functions*, SIAM Rev.**23**(1981), no. 2, 165–224. MR**618638**, https://doi.org/10.1137/1023037**[14]**Ezio Venturino,*On solving singular integral equations via a hyperbolic tangent quadrature rule*, Math. Comp.**47**(1986), no. 175, 159–167. MR**842128**, https://doi.org/10.1090/S0025-5718-1986-0842128-9**[15]**S. Welstead,*Orthogonal polynomials applied to the solution of singular integral equations*, Ph.D. Thesis, Purdue Univ., West Lafayette, IN, 1982.

Retrieve articles in *Mathematics of Computation*
with MSC:
65D30

Retrieve articles in all journals with MSC: 65D30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1990-1035926-3

Keywords:
Gauss quadratures,
Cauchy singular integral equations

Article copyright:
© Copyright 1990
American Mathematical Society