Application of Extrapolation Methods to Numerical Solution of Fredholm Integral Equations Related to Boundary Value Problems

Fredholm integral equations arise naturally in the context of ordinary and partial differential equations: Two-point boundary value problems can be reformulated as Fredholm integral equations, whose kernels are continuous but have finite jump discontinuities in their derivatives. Two-dimensional elliptic boundary problems can be reformulated as Fredholm integral equations with kernels that have singularities, some having logarithmic singularities. In this note, we describe quadrature methods whose accuracies can be improved at will. These are obtained by improving the underlying numerical quadrature formulas in a clever fashion. In the case of two-point boundary value problems, they are obtained by correcting the trapezoidal rule appropriately to the accuracy required. In the case of boundary integral equations, they are obtained by first correcting the basic trapezoidal rule and then extrapolating it to required accuracy.