# Technical Report CS-2019-01

 TR#: CS-2019-01 Class: CS Title: Novel Compact Numerical Quadrature Formulas for Supersingular Integrals of Periodic Functions Authors: Avram Sidi PDF Currently accessibly only within the Technion network Abstract: In the present work, we consider the numerical computation of finite-range supersingular integrals $I[f]=\intBar^b_a f(x)\,dx$, where $f(x)=g(x)/(x-t)^3$, that are defined in the sense of Hadamard Finite Part, assuming that $g\in C^\infty[a,b]$. We develop three numerical quadrature formulas of trapezoidal type for such integrals and analyze them. For the case in which $f(x)$ is also $T$-periodic with $T=b-a$, we modify these quadrature formulas in a suitable way; the resulting formulas read \begin{align*} \widehat{T}{}^{(0)}_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)}_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)}_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*} We pay particular attention to the case $f\in C^\infty(\mathbb{R}_t)$, $\mathbb{R}_t=\mathbb{R}\setminus\{t\pm kT\}^\infty_{k=0}$, in addition to being $T$-periodic, which arises in different contexts. For this case, we show that all three formulas have spectral'' accuracy; that is, $$\widehat{T}{}^{(s)}_n[f]-I[f]=O(n^{-\mu})\quad\text{as n\to\infty},\quad \forall \mu>0.$$ We provide a numerical example involving a periodic integrand that confirms our theoretical results. We also show how $\widehat{T}{}^{(2)}_n[f]$, which does not require derivative information, can be used for solving supersingular integral equations in an efficient manner. Copyright The above paper is copyright by the Technion, Author(s), or others. Please contact the author(s) for more information

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