PVTSI$^{\boldmath(m)}$: A Novel Approach to Computation of Hadamard Finite Parts of Nonperiodic Singular Integrals

Abstract

We consider the numerical computation of $I[f]=\intBar^b_a f(x)\,dx$, the Hadamard Finite Part of the finite-range singular integral ${\int }_a^{b}f(x)dx$, $f(x)=g(x)/(x-t{)}^m$ with $a<t<b$ and $m\in \{1,2,\dots \},$ assuming that (i)\,$g\in C^{\infty }(a,b)$ and (ii)\,$g(x)$ is allowed to have arbitrary integrable singularities at the endpoints $x=a$ and $x=b$. We first prove that $\intBar^b_a f(x)\,dx$ is invariant under any suitable variable transformation $x=\psi (\xi )$, $\psi :[\alpha ,\beta ]\to [a,b]$, hence there holds $\intBar^β_αF(ξ)\,dξ=\intBar^b_a f(x)\,dx$, where $F(\xi )=f(\psi (\xi )){\psi }^{\prime }(\xi )$. Based on this result, we next choose $\psi (\xi )$ such that the transformed integrand $F(\xi )$ is sufficiently periodic with period $\T=β-α$, and prove, with the help of some recent extension/generalization of the Euler--Maclaurin expansion, that we can apply to $\intBar^β_αF(ξ)\,dξ$ the quadrature formulas derived for periodic singular integrals developed in an earlier work of the author. We give a whole family of numerical quadrature formulas for $\intBar^β_αF(ξ)\,dξ$ for each $m$, which we denote ${\hat{T}}_{m,n}^{(s)}[\mathcal{F}]$, where $\mathcal{F}(\xi )$ is the $\T$-periodic extension of $F(\xi )$.

Number of pages: 32