Finite Part Integrals and Hypersingular Kernels

Transformation of hypersingular integrals and black-box cubature

In this paper, we will consider hypersingular integrals as they arise by transforming elliptic boundary value problems into boundary integral equations. First, local representations of these integrals will be derived. These representations contain so-called finite-part integrals. In the second step, these integrals are reformulated as improper integrals. We will show that these integrals can be...

Boundedness on Hardy-Sobolev Spaces for Hypersingular Marcinkiewicz Integrals with Variable Kernels

A Nonsingular Integral Formulation for the Helmholtz Eigenproblems of a Circular Domain

A nonsingular integral formulation for the Helmholtz eigenproblem is developed in this paper. This novel method contains only imaginary-part kernels instead of complex-part kernels in the complexvalued BEM. Based on the imaginary-part formulation without singular source, no singular or hypersingular integrals are present. Although this formulation avoids the computation of singular and hypersin...

Direct Evaluation of Hypersingular Galerkin Surface Integrals

A direct algorithm for evaluating hypersingular integrals arising in a three-dimensional Galerkin boundary integral analysis is presented. The singular integrals are defined as limits to the boundary, and by integrating two of the four dimensions analytically, the coincident integral is shown to be divergent. However, the divergent terms can be explicitly calculated and shown to cancel with cor...

Direct Evaluation of Hypersingular

A direct algorithm for evaluating hypersingular integrals arising in a three-dimensional Galerkin boundary integral analysis is presented. By integrating two of the four dimensions analytically, the coincident integration, deened as a limit to the boundary, is shown to be divergent. However, the divergent terms can be explicitly calculated and shown to cancel with corresponding singularities in...