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 $\int^b_a f(x)\,dx$, $f(x)=g(x)/(x-t)^{m}$ with $a<t<b$ and $m\in\{1,2,\ldots\},$ 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 $x=\psi(\xi)$, $\psi:[\alpha,\beta]\rightarrow[a,b]$, hence there holds $\intBar^\beta_\alpha F(\xi)\,d\xi=\intBar^b_a f(x)\,dx$, where $F(\xi)=f(\psi(\xi))\,\psi'(\xi)$. Based on this result, we next choose $\psi(\xi)$ such that the transformed integrand $F(\xi)$ is sufficiently periodic with period $\T=\beta-\alpha$, and prove, with the help of some recent extension/generalization of the Euler--Maclaurin expansion, that we can apply to $\intBar^\beta_\alpha F(\xi)\,d\xi$ 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 $\intBar^\beta_\alpha F(\xi)\,d\xi$ for each $m$, which we denote $\widehat{T}^{(s)}_{m,n}[{\cal F}]$, where ${\cal F}(\xi)$ is the $\T$-periodic extension of $F(\xi)$.