Hypersingular integrals at a corner

Abstract

For a smooth boundary, hypersingular integrals can be defined as a limit from the interior, the approach direction being taken, for convenience, normal to the surface. At a boundary corner, the limit process, with a necessarily non-normal approach direction, provides a valid definition of the hypersingular equation, as long as the same direction is employed for all integrations. The terms which are potentially singular in the limit are shown to cancel, provided the function approximations at the corner are consistent. The analytical formulas for the singular integrals are more complicated than for a smooth surface, but are easily obtained using symbolic computation. These techniques have been employed to accurately solve the ‘L-shaped domain’ potential considered by Jaswon and Symm ( Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, New York, USA, 1977).