A comprehensive approach to the approximate solution of singular integral equations

ABSTRACT. Both a collocation and a Galerkin method are described for the approximate solution of the complete singu­lar integral equation, with Cauchy principal value integral, defined on the arc ( — 1,1). Algorithms are described for all values of the index and an analysis of the discrete equations is given for each case. The approach is based on polynomial approximation over the arc ( — 1,1) and depends, in particular, upon the use of the Chebyshev polynomials of the first kind in the range space. The paper concludes with a convergence analysis which gives very satisfactory results. One surprising feature of the method is that except in the determination of the fundamental function, there is no need to evaluate any Cauchy principal value integrals. Furthermore, the method is not restricted to the particular cases of either constant or real coefficients and so is of wide applicability. 1. Introduction. In this paper singular integral equations of the form (1.1) M0 + A> = /, are considered where, if we let •»> W) - IA /' ^