Approximations of Hypersingular Integrals for Negative Fractional Exponent

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.