The numerical solution of first kind integral equations

In a recent paper, Babolian and Delves (hereafter BD) described a Chebyshev series method for the solution of first kind Fredholm integral equations. The method imposes regularisation constraints directly on the Chebyshev polynomial expansion of the solution, and involves two regularisation parameters, Cf and r. In this paper we develop a cross-validation algorithm capable of setting these parameters automatically. We show that the cross-validation scheme, coupled with the algorithm of BD leads to a stable regularised problem; and that the method can be implemented relatively inexpensively. Finally, we give a number of numerical examples showing that the method works well in practice.