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

Bialecki, Bernard; Stenger, Frank

doi:10.1090/S0025-5718-1988-0942147-X

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

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