PVTSI\(^{{{(m)}}}\): A novel approach to computation of Hadamard finite parts of nonperiodic singular integrals

We consider the numerical computation of \(I[f]=\int \!\!\!\!\!=^b_a f(x)\,dx\), 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 \(\int \!\!\!\!\!=^b_a f(x)\,dx\) is invariant under any legitimate variable transformation \(x=\psi (\xi )\), \(\psi :[\alpha ,\beta ]\rightarrow [a,b]\), hence there holds \(\int \!\!\!\!\!=^\beta _\alpha F(\xi )\,d\xi =\int \!\!\!\!\!=^b_a f(x)\,dx\), where \(F(\xi )=f(\psi (\xi ))\,\psi '(\xi )\). Based on this result, we next choose \(\psi (\xi )\) such that \(\mathcal{{F}}(\xi )\), the \({{\mathcal {T}}}\)-periodic extension of \(F(\xi )\), \({{\mathcal {T}}}=\beta -\alpha \), is sufficiently smooth, and prove, with the help of some recent extension/generalization of the Euler–Maclaurin expansion, that we can apply to \(\int \!\!\!\!\!=^\beta _\alpha F(\xi )\,d\xi \) the quadrature formulas derived for periodic singular integrals developed in an earlier work of the author: [A. Sidi, “Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions.” Calcolo, 58, 2021. Article number 22]. We give a whole family of numerical quadrature formulas for \(\int \!\!\!\!\!=^\beta _\alpha F(\xi )\,d\xi \) for each m, which we denote \({\widehat{T}}^{(s)}_{m,n}[\mathcal{{F}}]\). Letting \(G(\xi )=(\xi -\tau )^m F(\xi )\), with \(\tau \in (\alpha ,\beta )\) determined from \(t=\psi (\tau )\), and letting \(h={\mathcal {T}}/n\), for \(m=3\), for example, we have the three formulas We show that all of the formulas \({\widehat{T}}^{(s)}_{m,n}[\mathcal{{F}}]\) converge to I[f] as \(n\rightarrow \infty \); indeed, if \(\psi (\xi )\) is chosen such that \(\mathcal{{F}}^{(i)}(\alpha )=\mathcal{{F}}^{(i)}(\beta )=0\), \(i=0,1,\ldots ,q-1,\) and \(\mathcal{{F}}^{(q)}(\xi )\) is absolutely integrable in every closed interval not containing \(\xi =\tau \), then where q is a positive integer determined by the behavior of g(x) at \(x=a\) and \(x=b\) and also by \(\psi (\xi )\). As such, q can be increased arbitrarily (even to \(q=\infty \), thus inducing spectral convergence) by choosing \(\psi (\xi )\) suitably. We provide several numerical examples involving nonperiodic integrands and confirm our theoretical results.