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Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions. (English) Zbl 1472.65031

Calcolo 58, No. 2, Paper No. 22, 24 p. (2021); correction ibid. 58, No. 2, Paper No. 27, 1 p. (2021).
Summary: We consider the numerical computation of finite-range singular integrals \[ I[f]=\diagup\!\!\!\!\!\!{\int}^b_af(x)dx,\quad f(x)=\frac{g(x)}{(x-t)^m},\quad m=1,2,\dots,\quad a<t<b, \] 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 \(f\in C^\infty(\mathbb{R}_t)\), \(\mathbb{R}_t=\mathbb{R}\setminus\{t+ kT\}^\infty_{k=-\infty}\), \(T=b-a\). Using a generalization of the Euler-Maclaurin expansion developed in [A. Sidi, Math. Comput. 81, No. 280, 2159–2173 (2012; Zbl 1271.30011)], 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\limits^{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\limits^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\limits^n_{j=1}f(t+jh-h/2)- \frac{h}{2}\sum\limits^{2n}_{j=1}f(t+jh/2-h/4),\quad h=\frac{T}{n}. \end{align*} For all \(m\) and \(s\), we show that all of the numerical quadrature formulas \(\widehat{T}^{(s)}_{m,n}[f]\) have spectral accuracy; that is, \[ \widehat{T}^{(s)}_{m,n}[f]-I[f]=o(n^{-\mu})\quad \text{ as }\, {n\rightarrow \infty }\quad \forall \mu >0. \] We provide a numerical example involving a periodic integrand with \(m=3\) that confirms our convergence theory. We also show how the formulas \(\widehat{T}^{(s)}_{3,n}[f]\) can be used in an efficient manner for solving supersingular integral equations whose kernels have a \((x-t)^{-3}\) singularity. A similar approach can be applied for all \(m\).

MSC:

65D30 Numerical integration
41A55 Approximate quadratures
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)

Citations:

Zbl 1271.30011

Software:

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References:

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