HYPERSINGULAR AND FINITE-PART INTEGRALS IN THE BOUNDARY-ELEMENT METHOD

A new definition of the Hadamard finite part (HFP) of hypersingular integrals is proposed in this paper. This definition does not involve a limiting process. It is completely general and is valid for one as well as higher dimensional integrals, on closed as well as on open surfaces. It reduces, respectively, to the Cauchy principal value (CPV) and Riemann integral, respectively, for the special cases of strongly singular and weakly singular integrands. Of course, suitable symmetric exclusion zones must be chosen to realize CPV integrals. Starting with this new definition of the HFP of certain hypersingular boundary integral equations (HBIE) that arise in potential theory and in wave scattering, a regularization method is carried out in order to express the hypersingular integrals in terms of ones that are, at most, weakly singular. The regularized versions are completely consistent with those available in the recent literature where a different definition of the HFP was employed.