Bivariational Methods for Linear Integral Equations with Nonsymmetric Kernels

Abstract

Bivariational methods are used to develop upper and lower bounding functionals for arbitrary inner products $\langle {g,\phi } \rangle $ associated with the solutions $\phi $ of a class of linear integral equations with nonsymmetric kernels. The underlying structure of the variational principles is seen to relate to a linear combination of the integral equation and an adjoint equation. The new functionals derived are more accurate than previous bounds obtained from a direct approach. Their efficiency is illustrated by using them to calculate actual pointwise bounds on some solutions of integral equations. In two test cases, the accuracy achieved was superior to that arising from a standard 160-step Simpson quadrature calculation.