An Approach Based on Generalized Functions to Regularize Divergent Integrals

This article addresses weakly singular, hypersingular integrals, which arise when the boundary integral equation (BIE) methods are used for 3-D potential theory problem solutions. An approach based on the theory of distributions and the application of the second Green theorem has been explored for the calculation of such divergent integrals. The divergent integrals have been transformed to a form that allows easy and uniform calculation of weakly singular and hypersingular integrals. For flat boundary elements (BE), piecewise constants and piecewise linear approximations, only regular integrals over the contour of the BE have to be evaluated. Furthermore, all calculations can be done analytically, so no numerical integration is required. In the case of 3-D, rectangular and triangular BE have been considered. The behavior of divergent integrals has been studied in the context that the collocation point moves to the contour of the BE.