Technical Report CS-2019-01

Title: Novel Compact Numerical Quadrature Formulas for Supersingular Integrals of Periodic Functions
Authors: Avram Sidi
PDFCurrently 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.
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