PVTSI $^{\boldmath(m)}$：计算非周期奇异积分Hadamard有限部分的新方法
PVTSI$^{\boldmath(m)}$: A Novel Approach to Computation of Hadamard Finite Parts of Nonperiodic Singular Integrals
摘要：
本文考虑有限范围奇异积分$\int^b_a f(x)\,dx$的Hadamard有限部分$I[f]=\intBar^b_a f(x)\,dx$，$f(x)=g(x)/(x-t)^{m}$与$a<t<b$和$m\in\{1,2,\ldots\},$的数值计算，假设（i）$g\in C^\infty(a,b)$和（ii）$g(x)$在端点$x=a$和$x=b$上允许有任意可积奇点.我们首先证明了$\intBar^b_a f(x)\,dx$在任何合适的变量变换$x=ψ(ξ)$，$ψ:[α,β]\rightarrow[a,b]$下不变，因此存在$\intBar^β_αF(ξ)\,dξ=\intBar^b_a f(x)\,dx$，其中$F(ξ)=f(ψ(ξ))\,ψ'(ξ)$ .基于这一结果，我们接下来选择$ψ(ξ)$，使变换的被积函数$F(ξ)$是充分周期性的周期$\T=β-α$，并证明，与最近的一些扩展/推广的欧拉-麦克劳林展开的帮助下，我们可以适用于$\intBar^β_αF(ξ)\,dξ$的求积公式推导出的周期奇异积分在早期的工作的作者。对于每个$m$，我们给出了$\intBar^β_αF(ξ)\,dξ$的一族数值求积公式，我们表示为$\widehat{T}^{(s)}_{m,n}[{\cal F}]$，其中${\cal F}(ξ)$是$F(ξ)$的$\T$ -周期扩展。
英文摘要：
We consider the numerical computation of $I[f]=\intBar^b_a f(x)\,dx$, 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 $\intBar^b_a f(x)\,dx$ is invariant under any suitable variable transformation $x=ψ(ξ)$, $ψ:[α,β]\rightarrow[a,b]$, hence there holds $\intBar^β_αF(ξ)\,dξ=\intBar^b_a f(x)\,dx$, where $F(ξ)=f(ψ(ξ))\,ψ'(ξ)$. Based on this result, we next choose $ψ(ξ)$ such that the transformed integrand $F(ξ)$ is sufficiently periodic with period $\T=β-α$, and prove, with the help of some recent extension/generalization of the Euler--Maclaurin expansion, that we can apply to $\intBar^β_αF(ξ)\,dξ$ the quadrature formulas derived for periodic singular integrals developed in an earlier work of the author. We give a whole family of numerical quadrature formulas for $\intBar^β_αF(ξ)\,dξ$ for each $m$, which we denote $\widehat{T}^{(s)}_{m,n}[{\cal F}]$, where ${\cal F}(ξ)$ is the $\T$-periodic extension of $F(ξ)$.
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