Numerical Solution of Sigular Integral Equations Using Gauss-type Formulae I

Abstract

Of concern here are the Gauss—Chebyshev formulae for numerical solution of singular integral equations of the Cauchy-type. The coefficient matrix of the linear sustem of algebraic equations corresponding to the dominant term term is shown to have an inverse, which is expressed in a neat, closed form. Using the norm of the inverse matrix, the effect of quadrature errors on the computed solution is estimated. Using Jackson's theorems on “best approximation”, convergence of the procedure is proved under favourable conditions. In the course of analysis, a number of identities involving the zeros of Chebyshev polynomials of first and second kind is obtained.