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 - <p style="text-align: justify;">This article presents approximations of the hypersingular integrals&nbsp;$ʃ^b_ag(x)(x−t)^αdx$ and $ʃ^b_ag(x)|x−t|^αdx$ with arbitrary singular point $t ∈ (a, b)$ and negative fraction number $α &lt; −1$. 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.</p>