Asymptotic expansions of the error for hyper-singular integrals with an interval variable

Abstract

In this paper, we present high accuracy quadrature formulas for hyper-singular integrals $\int_{a}^{b}g(x)q^{\alpha}(x,t)\, dx$ , where $q(x,t)=|x-t|$ (or $x-t$ ), $t\in(a,b)$ , and $\alpha\leq-1$ (or $\alpha<-1$ ). If $g(x)$ is $2m+1$ times differentiable on $[a,b]$ , the asymptotic expansions of the error show that the convergence order is $O(h^{2\mu+1+\alpha})$ with $q(x,t)=|x-t|$ (or $x-t$ ) for $\alpha\leq-1$ (or $\alpha<-1$ and α being non-integer), and the error power is $O(h^{\eta})$ with $q(x,t)=x-t$ for α being integers less than −1, where $\eta =\min(2\mu,2\mu+2+\alpha)$ and $\mu=1,\ldots,m$ . Since the derivatives of the density function $g(x)$ in the quadrature formulas can be eliminated by means of the extrapolation method, the formulas can easily be 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.