Compact Numerical Quadrature Formulas for Hypersingular Integrals and Integral Equations

In the first part of this work, we derive compact numerical quadrature formulas for finite-range integrals \(I[f]=\int^{b}_{a}f(x)\,dx\), where f(x)=g(x)|x−t| ^β , β being real. Depending on the value of β, these integrals are defined either in the regular sense or in the sense of Hadamard finite part. Assuming that g∈C ^∞[a,b], or g∈C ^∞(a,b) but can have arbitrary algebraic singularities at x=a and/or x=b, and letting h=(b−a)/n, n an integer, we derive asymptotic expansions for \({T}^{*}_{n}[f]=h\sum_{1\leq j\leq n-1,\ x_{j}\neq t}f(x_{j})\), where x _j =a+jh and t∈{x ₁,…,x _n−1}. These asymptotic expansions are based on some recent generalizations of the Euler–Maclaurin expansion due to the author (A. Sidi, Euler–Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities, in Math. Comput., 2012), and are used to construct our quadrature formulas, whose accuracies are then increased at will by applying to them the Richardson extrapolation process. We pay particular attention to the case in which β=−2 and f(x) is T-periodic with T=b−a and \(f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}\), which arises in the context of periodic hypersingular integral equations. For this case, we propose the remarkably simple and compact quadrature formula \(\widehat{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)-\pi^{2} g(t)h^{-1}\), and show that \(\widehat{Q}_{n}[f]-I[f]=O(h^{\mu})\) as h→0 ∀μ>0, and that it is exact for a class of singular integrals involving trigonometric polynomials of degree at most n. We show how \(\widehat{Q}_{n}[f]\) can be used for solving hypersingular integral equations in an efficient manner. In the second part of this work, we derive the Euler–Maclaurin expansion for integrals \(I[f]=\int^{b}_{a} f(x)dx\), where f(x)=g(x)(x−t) ^β , with g(x) as before and β=−1,−3,−5,…, from which suitable quadrature formulas can be obtained. We revisit the case of β=−1, for which the known quadrature formula \(\widetilde{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)\) satisfies \(\widetilde{Q}_{n}[f]-I[f]=O(h^{\mu})\) as h→0 ∀μ>0, when f(x) is T-periodic with T=b−a and \(f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}\). We show that this formula too is exact for a class of singular integrals involving trigonometric polynomials of degree at most n−1. We provide numerical examples involving periodic integrands that confirm the theoretical results.