Product Formulas for Fredholm Integral Equations with Rational Kernel Functions

Abstract

In the last ten years the discretization of integrals with badly behaved kernel functions, on a finite interval of integration, has attracted the attention of several authors. The quadrature formulas proposed here are of interpolatory type and have the following form

$$\int_{ - 1}^1 {w\left( x \right)} K\left( {x,y} \right)f\left( x \right)dx \approx i = \sum\limits_{i = 1}^n {{w_{ni}}} \left( y \right)f\left( {{x_{ni}}} \right)$$

((1.1))

they are obtained by replacing f(x) by a polynomial, or by a piecewise polynomial function, which interpolates the function f(x) at the knots {x_ni}. The main feature of these rules, usually called product formulas, is that they integrate exactly the function w(x) K(x, y).

Work sponsored by the “Ministero della Pubblica Istruzione” of Italy.