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Asymptotic error expansions for hypersingular integrals

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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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Authors and Affiliations

  1. School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Jin Huang & Zhu Wang

  2. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

    Zhu Wang

  3. School of Mathematics, Sichuan University, Chengdu, 610064, China

    Rui Zhu

Authors
  1. Jin Huang
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  2. Zhu Wang
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  3. Rui Zhu
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Corresponding author

Correspondence to Zhu Wang.

Additional information

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

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