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

摘要

In this paper, we present a kind of quadrature rules for evaluating hyper-singular integrals ∫_a~b g(x)q~α(x, t)dx and ∫_a~b g(x)q~α(x, t) ln |x ? t|dx, where q(x, t) = |x ? t| (or x ? t), t ∈ (a, b) and α ≤ ?1(or α < ?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.