End-point behaviour of solutions to hypersingular integral equations

Abstract

We consider one-dimensional hypersingular integral equations over finite intervals; the integral must be interpreted as a finite-part integral. Such equations arise naturally in various physical situations, involving thin rigid bodies or cracks; examples are given. A method is developed for determining the behaviour of the solution to a hypersingular integral equation near the end-points of the interval of integration. The method uses the Mellin transform. Several examples are worked out in detail.