The convergence of a collocation method for a class of Cauchy singular integral equations

Abstract

In several recent papers we have studied the numerical solution of Cauchy singular integral equations of the first kind by a polynomial collocation method [4-61. This technique has proved to be quite efftcient for many problems, even when the kernel (but not the right-hand side) is discontinuous [4-61. The L, convergence and stability was established in [S] and [6]. In [ 121 Ioakimidis extended this method to solve Cauchy singular integral equations of the second kind with constant coefficients. Although some discussion of convergence was given it appears (at least to this author) that the proof is incomplete and the conditions on the kernel and the right-hand side are somewhat more restrictive than necessary. In this paper we will show that the convergence analysis given in [5,6] can be generalized to establish the L, convergence of collocation under rather mild restrictions on the data. Although a variety of numerical techniques such as Galerkin’s method and direct quadrature have been used extensively to solve Cauchy singular equations, they seem to be generally suited for problems with smooth kernels. For problems with discontinuous kernels collocation methods appear to be a reasonable compromise between time-consuming Galerkin methods [2] and direct quadrature methods which require the evaluation of the kernel at pairs of points where the kernel may become unbounded [3, 131. It is well known that for Fredholm equations collocation is related to product integration quadrature methods [ 11. Since such methods seem not to have been developed for the solution of Cauchy singular equations collocation appears to be a good way of dealing with fairly general integral equations of this type [4, 71.