Euler–Maclaurin expansions for integrals with endpoint singularities: a new perspective (original) (raw)
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Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities
Constructive Approximation, 2012
ABSTRACT In this paper, we provide the Euler–Maclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals I[f]=òbaf(x)dxI[f]=\int^{b}_{a}f(x)\,dx, where f∈C ∞(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms [^(P)](y),Ps(y)\widehat{P}(y),P_{s}(y) and [^(Q)](y),Qs(y)\widehat{Q}(y),Q_{s}(y) are polynomials in y. The γ s and δ s are distinct, complex in general, and different from −1. They also satisfy [^(P)](y) º 0\widehat{P}(y)\equiv0 and [^(Q)](y) º 0\widehat{Q}(y)\equiv0. They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x→a+ and x→b−. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let Dw=\fracddwD_{\omega}=\frac{d}{d\omega}, h=(b−a)/n, where n is a positive integer, and define \checkTn[f]=hån-1i=1f(a+ih)\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih). Then with [^(P)](y)=å[^(p)]i=0[^(c)]iyi\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i} and [^(Q)](y)=å[^(q)]i=0[^(d)]iyi\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}, one of these results reads si = limn®¥[ånk=1\frac(logk)ik-\frac(logn)i+1i+1]\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}], i=0,1,… . KeywordsEuler–Maclaurin expansions–Asymptotic expansions–Trapezoidal rule–Endpoint singularities–Algebraic singularities–Logarithmic singularities–Hadamard finite part–Zeta function–Stieltjes constants
SUMMARY In the recent works (Commun. Numer. Meth. Engng 2001; 17:881; to appear), the superiority of the non-linear transformations containing a real parameter b = 0 has been demonstrated in numerical evaluation of weakly singular integrals. Based on these transformations, we define a so-called parametric sigmoidal transformation and employ it to evaluate the Cauchy principal value and Hadamard finite-part integrals by using the Euler–Maclaurin formula. Better approximation is expected due to the prominent properties of the parametric sigmoidal transformation of whose local behaviour near x = 0 is governed by parameter b. Through the asymptotic error analysis of the Euler–Maclaurin formula using the parametric sigmoidal transformation, we can observe that it provides a distinct improvement on its predecessors using traditional sigmoidal transformations. Numerical results of some examples show the availability of the present method.
2021
Let I[f] = ⨎ b a f(x) dx, f(x) = g(x) (x − t)m , m = 1, 2, . . . , a < t < b, assuming that g ∈ C∞[a, b] such that f(x) is T -periodic, T = b−a, and f(x) ∈ C∞(Rt) with Rt = R\{t+kT}∞k=−∞. Here ⨎a f(x) dx stands for the Hadamard Finite Part (HFP) of the singular integral ∫ b a f(x) dx that does not exist in the regular sense. In a recent work, we unified the treatment of these HFP integrals by using a generalization of the Euler–Maclaurin expansion due to the author and developed a number of numerical quadrature formulas T̂ (s) m,n[f] of trapezoidal type for I[f] for all m. For example, three numerical quadrature formulas of trapezoidal type result from this approach for the case m = 3, and these are T̂ (0) 3,n[f] = h n−1 ∑ j=1 f(t + jh) − π 3 g ′(t)h−1 + 1 6 g ′′′(t)h, h = Tn , T̂ (1) 3,n[f] = h n ∑ j=1 f(t + jh − h/2) − π g′(t)h−1, h = Tn , T̂ (2) 3,n[f] = 2h n ∑ j=1 f(t + jh − h/2) − h 2 2n ∑ j=1 f(t + jh/2 − h/4), h = Tn . For all m and s, we showed that all of the numerica...
2021
We consider the numerical computation of I[f] = ⨎ba f(x) dx, the Hadamard Finite Part of the finite-range singular integral ∫ b a f(x) dx, f(x) = g(x)/(x − t) m with a < t < b and m ∈ {1, 2, . . .}, assuming that (i) g ∈ C(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 ⨎ba f(x) dx is invariant under any suitable variable transformation x = ψ(ξ), ψ ∶ [α, β] → [a, b], hence there holds ⨎βα F (ξ) dξ = ⨎ 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 ⨎βα F (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 ⨎βα F (ξ) dξ for each m, which w...
Journal of Mathematical Inequalities, 2009
Some integral remainders of Trapezoidal, Corrective Trapezoidal and Simpson formulas are given. The Hölder inequality and Henstock-Kurzweil integral are used to estimate the remainders in terms of Alexiewicz and Lebesgue p -norms. Mathematics subject classification (2000): Primary 28B05, 46G10 ; Secondary 26A39. 243 244 XIAOFENG DING, GUOJU YE AND WEI-CHI YANG 2. Preliminaries Let [a, b] be a compact interval in R, μ stand for the Lebesgue measure. A partial partition D in [a, b] is a finite collection of interval-point pairs (I, ξ ) with non-overlapping intervals I ⊂ [a, b], ξ ∈ [a, b] being the associated point of I . We write D = {(I, ξ )} . A partial partition D = {(I, ξ )} in [a, b] is a partition of [a, b] if the union of all the intervals I from D equals [a, b].