Approximations of Hypersingular Integrals for Negative Fractional Exponent (Journal of Computational Mathematics, 2018)
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 otherimportant ones which have not been addressed yet. The corresponding mid-rectangularformulas and extrapolations, which can be calculated in fairly straightforward ways, areinvestigated. Numerical examples are provided to illustrate the features of the numericalmethods and verify the theoretical conclusions.