Euler–Maclaurin Expansions and Quadrature Formulas of Hyper-singular Integrals with an Interval Variable

Abstract

In this paper, we present a kind of quadrature rules for evaluating hyper-singular integrals $\int _{a}^{b}g(x)q^{\alpha }(x,t)dx$ and $\int _{a}^{b}g(x)q^{\alpha }(x,t)\ln |x-t| dx$, where $q(x,t)=|x-t|$ (or $x-t$), $t\in (a,b)$ and $\alpha \le -1$(or $\alpha <-1$). Since the derivatives of density function g(x) in the quadrature formulas can be eliminated by means of the extrapolation method, the formulas can be easily applied to solving corresponding hyper-singular boundary integral equations. The reliability and efficiency of the proposed formulas in this paper are demonstrated by some numerical examples.