Asymptotic error expansions for hypersingular integrals

作者：Huang, Jin; Wang, Zhu^*; Zhu, Rui

来源：Advances in Computational Mathematics, 2013, 38(2): 257-279.

DOI：10.1007/s10444-011-9236-x

• Hypersingular integral
• Asymptotic expansion
• Quadrature formula
• Richardson extrapolation

This paper presents quadrature formulae for hypersingular integrals , where a < t < b and 0 < alpha a parts per thousand currency signaEuro parts per thousand 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 (2 mu) ) for alpha = 1 and O(h (2 mu -aEuro parts per thousand alpha) ) for 0 < alpha < 1, where mu 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.

• 出版日期2013-2
• 单位电子科技大学; 四川大学