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 integralsis 2m + 1 times differentiable on [a, b], the asymptotic expansions of the error show that the convergence order is O(h 2μ+1+α ) with q(x, t) = |x -t| (or x -t) for α ≤ -1 (or α < -1 and α being non-integer), and the error power is O(h η ) with q(x, t) = x -t for α being integers less than -1, where η = min(2μ, 2μ + 2 + α) and μ = 1, . . . , 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. MSC: 45E99; 65D30; 65D32; 41A55