## Sinc-Nyström method for numerical solution of one-dimensional Cauchy singular integral equation given on a smooth arc in the complex plane

HTML articles powered by AMS MathViewer

- by Bernard Bialecki and Frank Stenger PDF
- Math. Comp.
**51**(1988), 133-165 Request permission

## Abstract:

We develop a numerical method based on Sinc functions to obtain an approximate solution of a one-dimensional Cauchy singular integral equation (CSIE) over an arbitrary, smooth, open arc*L*of finite length in the complex plane. At the outset, we reduce the CSIE to a Fredholm integral equation of the second kind via a regularization procedure. We then obtain an approximate solution to the Fredholm integral equation by means of Nyström’s method based on a Sinc quadrature rule. We approximate the matrix and right-hand side of the resulting linear system by an efficient method of computing the Cauchy principal value integrals. The error of an

*N*-point approximation converges to zero at the rate $O({e^{ - c{N^{1/2}}}})$, as $N \to \infty$, provided that the coefficients of the CSIE are analytic in a region

*D*containing the arc

*L*and satisfy a Lipschitz condition in

*D*.

## References

- Philip M. Anselone,
*Collectively compact operator approximation theory and applications to integral equations*, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. With an appendix by Joel Davis. MR**0443383** - Kendall E. Atkinson,
*A survey of numerical methods for the solution of Fredholm integral equations of the second kind*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. MR**0483585** - M. L. Dow and David Elliott,
*The numerical solution of singular integral equations over $(-1,\,1)$*, SIAM J. Numer. Anal.**16**(1979), no. 1, 115–134. MR**518688**, DOI 10.1137/0716009 - David Elliott,
*The classical collocation method for singular integral equations*, SIAM J. Numer. Anal.**19**(1982), no. 4, 816–832. MR**664887**, DOI 10.1137/0719057 - David Elliott,
*Rates of convergence for the method of classical collocation for solving singular integral equations*, SIAM J. Numer. Anal.**21**(1984), no. 1, 136–148. MR**731218**, DOI 10.1137/0721009
D. Elliott & F. Stenger, - G. M. Goluzin,
*Geometric theory of functions of a complex variable*, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR**0247039**, DOI 10.1090/mmono/026 - N. I. Ioakimidis,
*On the weighted Galerkin method of numerical solution of Cauchy type singular integral equations*, SIAM J. Numer. Anal.**18**(1981), no. 6, 1120–1127. MR**639002**, DOI 10.1137/0718076 - N. I. Ioakimidis,
*Further convergence results for the weighted Galerkin method of numerical solution of Cauchy-type singular integral equations*, Math. Comp.**41**(1983), no. 163, 79–85. MR**701625**, DOI 10.1090/S0025-5718-1983-0701625-4 - Peter Linz,
*An analysis of a method for solving singular integral equations*, Nordisk Tidskr. Informationsbehandling (BIT)**17**(1977), no. 3, 329–337. MR**483594**, DOI 10.1007/bf01932153 - George Miel,
*On the Galerkin and collocation methods for a Cauchy singular integral equation*, SIAM J. Numer. Anal.**23**(1986), no. 1, 135–143. MR**821910**, DOI 10.1137/0723009 - 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** - Frank Stenger,
*Numerical methods based on Whittaker cardinal, or sinc functions*, SIAM Rev.**23**(1981), no. 2, 165–224. MR**618638**, DOI 10.1137/1023037 - K. S. Thomas,
*Galerkin methods for singular integral equations*, Math. Comp.**36**(1981), no. 153, 193–205. MR**595052**, DOI 10.1090/S0025-5718-1981-0595052-9 - J. H. Wilkinson,
*Rounding errors in algebraic processes*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR**0161456**

*Sinc Method of Solution of Singular Integral Equations*, IMACS Symposium on Numerical Solution of Singular Integral Equations, IMACS, 1984, pp. 27-35.

## Additional Information

- © Copyright 1988 American Mathematical Society
- Journal: Math. Comp.
**51**(1988), 133-165 - MSC: Primary 65R20
- DOI: https://doi.org/10.1090/S0025-5718-1988-0942147-X
- MathSciNet review: 942147