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

TL;DR

This paper introduces PVTSI$^{(m)}$, a new method for numerically computing Hadamard finite parts of nonperiodic singular integrals by transforming them into periodic integrals and applying specialized quadrature formulas.

Contribution

The paper proves invariance of Hadamard finite parts under variable transformation and develops a family of quadrature formulas for periodic singular integrals based on this invariance.

Findings

Proved invariance of Hadamard finite parts under variable change.

Constructed quadrature formulas for periodic singular integrals.

Validated the effectiveness of the new numerical approach.

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 $F(\xi)$.