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Volume 36, Issue 5
Approximations of Hypersingular Integrals for Negative Fractional Exponent

Chaolang Hu & Tao Lü

DOI: 10.4208/jcm.1703-m2016-0544

J. Comp. Math., 36 (2018), pp. 627-643.

Published online: 2018-06

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  • Abstract

This article presents approximations of the hypersingular integrals $ʃ^b_ag(x)(x−t)^αdx$ and $ʃ^b_ag(x)|x−t|^αdx$ with arbitrary singular point $t ∈ (a, b)$ and negative fraction number $α < −1$. These general expansions are applicable to a large range of hypersingular integrals, including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet. The corresponding mid-rectangular formulas and extrapolations, which can be calculated in fairly straightforward ways, are investigated. Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions.

  • Keywords

Hypersingular integral, Negative fractional exponent, Mid-rectangular quadrature formula, Extrapolation.

  • AMS Subject Headings

65B15, 42B20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

huchaolang@scu.edu.cn (Chaolang Hu)

lutao1940@sina.com (Tao Lü)

  • BibTex
  • RIS
  • TXT
@Article{JCM-36-627, author = {Hu , Chaolang and Lü , Tao}, title = {Approximations of Hypersingular Integrals for Negative Fractional Exponent}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {5}, pages = {627--643}, abstract = {

This article presents approximations of the hypersingular integrals $ʃ^b_ag(x)(x−t)^αdx$ and $ʃ^b_ag(x)|x−t|^αdx$ with arbitrary singular point $t ∈ (a, b)$ and negative fraction number $α < −1$. These general expansions are applicable to a large range of hypersingular integrals, including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet. The corresponding mid-rectangular formulas and extrapolations, which can be calculated in fairly straightforward ways, are investigated. Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1703-m2016-0544}, url = {http://global-sci.org/intro/article_detail/jcm/12449.html} }
TY - JOUR T1 - Approximations of Hypersingular Integrals for Negative Fractional Exponent AU - Hu , Chaolang AU - Lü , Tao JO - Journal of Computational Mathematics VL - 5 SP - 627 EP - 643 PY - 2018 DA - 2018/06 SN - 36 DO - http://doi.org/10.4208/jcm.1703-m2016-0544 UR - https://global-sci.org/intro/article_detail/jcm/12449.html KW - Hypersingular integral, Negative fractional exponent, Mid-rectangular quadrature formula, Extrapolation. AB -

This article presents approximations of the hypersingular integrals $ʃ^b_ag(x)(x−t)^αdx$ and $ʃ^b_ag(x)|x−t|^αdx$ with arbitrary singular point $t ∈ (a, b)$ and negative fraction number $α < −1$. These general expansions are applicable to a large range of hypersingular integrals, including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet. The corresponding mid-rectangular formulas and extrapolations, which can be calculated in fairly straightforward ways, are investigated. Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions.

Chaolang Hu & Tao Lü. (2020). Approximations of Hypersingular Integrals for Negative Fractional Exponent. Journal of Computational Mathematics. 36 (5). 627-643. doi:10.4208/jcm.1703-m2016-0544
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