On the free terms of the dual BIE for N-dimensional Laplace problems

Abstract

Dual boundary integral equations for the N-dimensional Laplace problems with a smooth boundary are derived by using the contour approach surrounding the singularity. The potentials resulted from the four kernel functions in the dual formulation have different properties across the smooth boundary. For the generalization, we focus on the N-dimensional Laplace equation. The Hadamard principal value (H.P.V.) is derived naturally and is composed of two parts, the Cauchy principal value (C.P.V.) and an unbounded boundary term. The hypersingular integral is not a divergent integral since we can collect the C.P.V. and the unbounded term together. Besides, the weighting of the free term contributed by different kernels is also examined. Finally, a special case of the four-dimensional Laplace equation is implemented and the free term, for any dimension are obtained. The contributions of the free terms for the boundary normal derivative of potential due to the single (L kernel) and the double (M kernel) layer potentials are 1/N and (N-1)/N, respectively. It is an interesting phenomenon that the hypersingular kernel contributes more than the singular kernel, and, in addition, the former also yields an unbounded boundary term.