Calcolo ( IF 1.4 ) Pub Date : 2021-05-06 , DOI: 10.1007/s10092-021-00407-8 Avram Sidi
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We consider the numerical computation of finite-range singular integrals
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, Euler–Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp., 81:2159–2173, 2012], 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
$$\begin{aligned} {\widehat{T}}^{(0)}_{3,n}[f]&=h\sum ^{n-1}_{j=1}f(t+jh)-\frac{\pi ^2}{3}\,g'(t)\,h^{-1} +\frac{1}{6}\,g'''(t)\,h, \quad h=\frac{T}{n},\\ {\widehat{T}}^{(1)}_{3,n}[f]&=h\sum ^n_{j=1}f(t+jh-h/2)-\pi ^2\,g'(t)\,h^{-1},\quad h=\frac{T}{n},\\ {\widehat{T}}^{(2)}_{3,n}[f]&=2h\sum ^n_{j=1}f(t+jh-h/2)- \frac{h}{2}\sum ^{2n}_{j=1}f(t+jh/2-h/4),\quad h=\frac{T}{n}. \end{aligned}$$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,
$$\begin{aligned} {\widehat{T}}^{(s)}_{m,n}[f]-I[f]=o(n^{-\mu })\quad \text {as}\, {n\rightarrow \infty }\quad \forall \mu >0. \end{aligned}$$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.
中文翻译：
周期函数奇异积分的Hadamard有限部分的统一紧致数值正交公式
我们考虑有限范围奇异积分的数值计算
在Hadamard有限零件的意义上定义，假设\（g \ in C ^ \ infty [a，b] \）和\（f（x）\ in C ^ \ infty（{\ mathbb {R}} _t）\）为T周期，\（f \ in C ^ \ infty（{\ mathbb {R}} _ t），\） \（{\ mathbb {R}} _ t = {\ mathbb {R}} { \ setminus} \ {t + kT \} ^ \ infty _ {k =-\ infty} \），\（T = ba \）。使用[A.开发的Euler-Maclaurin展开的一般化。具有任意代数端点奇点的积分的Sidi，Euler–Maclaurin展开。数学。比较 ，81：2159–2173，2012]，我们统一了对这些积分的处理。对于每个m，我们开发出许多数字正交公式\（{\ widehat {T}} ^ {（s}} _ {m，n} [f] \）I [ f ]的梯形类型。例如，对于情况\（m = 3 \），此方法得出了梯形类型的三个数值正交公式，它们是
$$ \ begin {aligned} {\ widehat {T}} ^ {（0）} _ {3，n} [f]＆= h \ sum ^ {n-1} _ {j = 1} f（t + jh）-\ frac {\ pi ^ 2} {3} \，g'（t）\，h ^ {-1} + \ frac {1} {6} \，g'''（t）\，h ，\ quad h = \ frac {T} {n}，\\ {\ widehat {T}} ^ {（1）} _ {3，n} [f]＆= h \ sum ^ n_ {j = 1} f（t + jh-h / 2）-\ pi ^ 2 \，g'（t）\，h ^ {-1}，\ quad h = \ frac {T} {n}，\\ {\ widehat { T}} ^ {（2）} _ {3，n} [f]＆= 2h \ sum ^ n_ {j = 1} f（t + jh-h / 2）-\ frac {h} {2} \和^ {2n} _ {j = 1} f（t + jh / 2-h / 4），\ quad h = \ frac {T} {n}。\ end {aligned} $$对于所有m和s，我们证明所有数值正交公式\（{\ widehat {T}} ^ {（s）} _ {m，n} [f] \）具有光谱精度；那是，
$$ \ begin {aligned} {\ widehat {T}} ^ {{s}} _ {m，n} [f] -I [f] = o（n ^ {-\ mu}）\ quad \ text { }}，{n \ rightarrow \ infty} \ quad \ forall \ mu> 0。\ end {aligned} $$我们提供了一个包含\（m = 3 \）的周期被积数的数值示例，这证实了我们的收敛理论。我们还将展示如何有效地使用公式\（{\ widehat {T}} {} ^ {（s} _ {3，n} [f] \）来求解内核为a的超奇异积分方程。\（（xt）^ {-3} \）奇异。可以对所有m应用类似的方法。
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