Accelerated degenerate-kernel methods for linear integral equations

A new approach to the theory of kernel approximations is developed for the numerical solution of Fredholm integral equations of the second kind. The objective is to determine the solution to any desired accuracy using a degenerate-kernel operator of fixed rank — that is, without increasing the number of terms in the approximate kernel. Consequently, high accuracy is achieved without incurring the computational costs associated with solving large algebraic systems. The procedure employs projection techniques to construct accelerated degenerate-kernel schemes that give rise to a bisequence of higher-order approximate solutions. Error expressions are obtained and a numerical example of solving a weakly singular equation is given.