Asymptotic error expansions for hypersingular integrals

This paper presents quadrature formulae for hypersingular integrals $int_a^bfrac{g(x)}{|x-t|^{1+alpha }}mathrm{d}x$ , where a?t?b and 0?α?≤?1. The asymptotic error estimates obtained by Euler–Maclaurin expansions show that, if g(x) is 2m times differentiable on [a,b], the order of convergence is O(h ^2μ ) for α?=?1 and O(h ^2μ???α ) for 0?α?μ is a positive integer determined by the integrand. The advantages of these formulae are as follows: (1) using the formulae to evaluate hypersingular integrals is straightforward without need of calculating any weight; (2) the quadratures only involve g(x), but not its derivatives, which implies these formulae can be easily applied for solving corresponding hypersingular boundary integral equations in that g(x) is unknown; (3) more precise quadratures can be obtained by the Richardson extrapolation. Numerical experiments in this paper verify the theoretical analysis.