In this paper the author gives a new proof to evaluate the convergence of a collocation method for the Abel integral equations of the first kind.

The authors consider some procedure for solving a class of singular integral equations of Cauchy type by the collocation method on the zeros of orthogonal polynomials with respect to a weight function similar to that of Jacobi. Uniform convergence theorems are proved.

A numerical method to solve Abel-type integral equations of first kind is given. In this paper we suggest the research of a numerical solution for Abel-type integral equations of the first kind, by using a collocation method employing an interpolatory product-quadrature formula with a trigonometric polynomial of the first order. Some results of numerical examples are reported.