Unified Compact Numerical Quadrature Formulas for Hadamard Finite Parts of Singular Integrals of Periodic Functions

TL;DR

This paper develops unified numerical quadrature formulas of trapezoidal type for computing Hadamard finite part integrals of periodic functions with algebraic singularities, improving accuracy and efficiency.

Contribution

It introduces a generalized Euler–Maclaurin expansion to unify and extend quadrature formulas for singular integrals of periodic functions.

Findings

Derived multiple trapezoidal-type quadrature formulas for singular integrals.

Formulas explicitly incorporate derivatives of the smooth part of the integrand.

Enhanced numerical methods for accurate computation of Hadamard finite part integrals.

Abstract

We consider the numerical computation of finite-range singular integrals $$I[f]=\intBar^b_a f(x)\,dx,\quad f(x)=\frac{g(x)}{(x-t)^m},\quad m=1,2,\ldots,\quad a<t<b,$$ that are defined in the sense of Hadamard Finite Part, assuming that $g\in C^\infty[a,b]$ and $f(x)\in C^\infty(\mathbb{R}_t)$ is $T$-periodic with $\mathbb{R}_t=\mathbb{R}\setminus\{t+ kT\}^\infty_{k=-\infty}$, $T=b-a$. Using a generalization of the Euler--Maclaurin expansion developed in [A. Sidi, {Euler--Maclaurin} expansions for integrals with arbitrary algebraic endpoint singularities. {\em Math. Comp.}, 81:2159--2173, 2012], we unify the treatment of these integrals. For each $m$, we develop a number of numerical quadrature formulas $\widehat{T}^{(s)}_{m,n}[f]$ of trapezoidal type for $I[f]$. For example, three numerical quadrature formulas of trapezoidal type result from this approach for the case $m=3$, and these are \begin{align*} \widehat{T}^{(0)}_{3,n}[f]&=h\sum^{n-1}_{j=1}f(t+jh)-\frac{\pi^2}{3}\,g'(t)\,h^{-1} +\frac{1}{6}\,g'''(t)\,h, \quad h=\frac{T}{n}, \widehat{T}^{(1)}_{3,n}[f]&=h\sum^n_{j=1}f(t+jh-h/2)-\pi^2\,g'(t)\,h^{-1},\quad h=\frac{T}{n}, \widehat{T}^{(2)}_{3,n}[f]&=2h\sum^n_{j=1}f(t+jh-h/2)- \frac{h}{2}\sum^{2n}_{j=1}f(t+jh/2-h/4),\quad h=\frac{T}{n}.\end{align*}