Asymptotic error expansions for hypersingular integrals

• DOI: 10.1007/s10444-011-9236-x
• Publication Date: 2011-09-29T02:05:09Z
• Authors: Jin Huang; Zhu Wang; Rui Zhu
• Keywords: Quadrature (astronomy); Richardson extrapolation

This paper presents quadrature formulae for hypersingular integrals $\int_a^b\frac{g(x)}{|x-t|^{1+\alpha }}\mathrm{d}x$ , where a?<?t?<?b and 0?<?????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 ??=?1 and O(h 2μ???? ) for 0?<???<?1, where 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.