Nyström’s Iterative Variant Methods for the Solution of Cauchy Singular Integral Equations

Abstract

Iterative methods for the solution of algebraic systems and integral equations are becoming increasingly popular, because they can take advantage of the speed of vector and parallel computers. In this paper, Nystrom’s iterative variant methods developed for the solution of Fredholm integral equations are extended to the solution of Cauchy singular integral equations. Gauss–Chebyshev and Lobatto–Chebyshev quadrature approximations are used to define two iterative algorithms, GIV and LIV, and their convergence for all continuously differentiable kernels is proven. For arbitrary kernels the complexity is $O(m^2 )$ operations per iteration step, where m is the number of node points. However, for certain kernels the complexity can be reduced to $O(m\log _2 m)$ by using a fast summation algorithm. An extrapolation technique based on the average of the Gauss–Chebyshev and Lobatto–Chebyshev methods is also introduced. The average is usually as accurate as the Lobatto–Chebyshev or Gauss–Chebyshev method with twice...