Three algorithms for Hadamard finite-part integrals and fractional derivatives
type: Item
instance of: scholarly article
title: Three algorithms for Hadamard finite-part integrals and fractional derivatives
zbMATH Open document ID: 0858.65019
DOI: 10.1016/0377-0427(94)00113-5
published in: Journal of Computational and Applied Mathematics
publication date: 1997-03-25T00:00:00Z
review text: To evaluate the Hadamard finite part integral of the form \[ \int^{+1}_{-1} \bigl(f(t)/(1-t)^{1+ \alpha} \bigr)dt, \] \(\alpha\) a positive noninteger, the author develops three algorithms. The algorithms are based on the Chebyshev series expansion of \(f\), on interpolations to \(f\) at the zeros of Chebyshev polynomials of the first kind and again interpolation at these zeros under the additional assumption that \(f^{(k)}(-1) = f^{(k)} (1)=0\) for \(k=0,\dots,n\), \(n<\alpha <n+1\). The latter condition can always be achieved by subtracting from \(f \in C^{(n+1)} [-1,+1]\) an appropriate polynomial and enables the author to give a convergence proof without assuming that \(f\) is smoother than it needs to be for the Hadamard integral to exist. Convergence of the three algorithms is proved and numerical examples are given. Reviewer's annotation: The reader will encounter several formulas of classical special functions, which are utilized skilfully in the derivation of the rules and in the convergence proofs.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 903478
zbMATH Keywords: fractional derivatives
zbMATH Keywords: polynomial interpolation
zbMATH Keywords: stability
zbMATH Keywords: Hadamard finite part integral
zbMATH Keywords: algorithms
zbMATH Keywords: Chebyshev series expansion
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: convergence
zbMATH Keywords: numerical examples
author: David Elliott
MaRDI profile type: Publication
full work available at URL: 0377-0427(94)00113-5
OpenAlex ID: W1994511284
cites work: Q3661894
cites work: Q5721434
cites work: On the approximate evaluation of Hadamard finite-part integrals
cites work: The Evaluation and Estimation of the Coefficients in the Chebyshev Series Expansion of a Function
cites work: An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals
cites work: Some Estimates of the Coefficients in the Chebyshev Series Expansion of a Function
cites work: Q5824493
cites work: On the convergence of polynomials interpolating at the zeroes ofT n (x)
cites work: An Elementary Simultaneous Approximation Theorem
cites work: Q4092856
cites work: The special functions and their approximations. Vol. I, II
cites work: On Lagrange and Hermite interpolation. I
cites work: Q3864730
cites work: A quadrature rule for finite-part integrals
cites work: The evaluation and application of some modified moments
cites work: Calculating singular integrals as an ill-posed problem
cites work: Q3997537
cites work: Q3248306
Wikidata QID: Q127939658
description: scientific article; zbMATH DE number 903478