An approximate method for evaluating hypersingular integrals

Abstract

This paper describes a new method for direct numerical evaluation of multidimensional hypersingular integrals assigned on smooth curves and surfaces. These integrals arise when the boundary integral equations are used to solve problems of mechanics, electrodynamics, aerodynamics, etc. The hypersingular integrals are considered, in the sense of Hadamard, as finite parts. The main advantage of the proposed method is the numerical computation of the hypersingular integrals by the direct application of the developed cubature formulas, thus requiring little analytical pre-work. The method is not restricted to the type of problem however and may be easily applied to any hypersingular integrals. The convergence of the proposed technique has been proved and error estimates are given. An illustrative example demonstrates the accuracy and efficiency of the method.