Numerical evaluation of certain strongly singular integrals

Abstract

Interpolatory quadrature rules have been well studied and widely applied for numerical integration in the presence of simple singularities. However, these rules have been used less for strongly singular integrals due to the fact that divergence and instability problems increase with respect to the order of the singularity. In the present paper we study this aspect proving convergence and stability of suitable interpolatory rules to compute certain singular and hypersingular integrals. The particular case of these integrals for oscillatory functions is also examined. Some numerical tests confirming our theoretical analysis are also presented.