Abstract
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 < 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 < 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.
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Communicated by Yuesheng Xu.
The work is supported by National Science Foundation of China (10871034). The second author was supported in part by NSF grants DMS-1016450 and AFOSR grant FA9550-08-1-0136.
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Huang, J., Wang, Z. & Zhu, R. Asymptotic error expansions for hypersingular integrals. Adv Comput Math 38, 257–279 (2013). https://doi.org/10.1007/s10444-011-9236-x
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DOI: https://doi.org/10.1007/s10444-011-9236-x