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 \( \int _{a}^{b}g(x)q^{\alpha }(x,t)dx\) 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.

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