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

In this paper, we present a kind of quadrature rules for evaluating hyper-singular integrals integral(b)(a) g(x)q(alpha)(x, t) dx and integral(b)(a) g(x)q(alpha)(x, t) ln |x - t| dx, where q( x, t) = |x - t| (or x - t), t is an element of(a, b) and alpha <= -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.