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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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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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