Weighted nonstandard quadrature formulas based on values of linear differential operators
type: Item
instance of: scholarly article
title: Weighted nonstandard quadrature formulas based on values of linear differential operators
zbMATH Open document ID: 1490.65040
author: Mohammad Masjed-Jamei
author: Zahra Moalemi
author: Gradimir V. Milovanović
published in: Journal of Computational and Applied Mathematics
publication date: 2022-03-15T00:00:00Z
review text: The paper starts with a literature survey of several extensions and variations of Gaussian quadrature formulas that make use of values for the integrand and its (higher order) derivatives in the free and/or prefixed quadrature nodes and still keep a maximal polynomial degree of exactness. The main contribution is the construction of a quadrature formula with real nodes of the form \(\int_a^b f(x)w(x)dx\approx\sum_{k=1}^n w_k \mathcal{L}[f](x_k)\) where \(\mathcal{L}[f](x)=A(x)f(x)+B(x)f'(x)\) is a differential operator where \(A\) and \(B\) are predefined continuous functions and \(B(x)>0\) in \([a,b]\). The integrand is continuously differentiable and vanishes at \(\lambda\in\mathbb{R}\). Note that \(\lambda\) need not be in \([a,b]\) but continuity conditions need to hold in the interval that contains both \([a,b]\) and \(\lambda\). It is proved how the nodes and the weights can be computed such that the quadrature has algebraic degree of exactness \(2n\). It boils down to finding nodes and weights for a Gaussian quadrature formula for an auxiliary weight. The latter is characterized by its moments. Several numerical examples based on Mathematica software illustrate the result.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65D30
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 7489833
zbMATH Keywords: Gaussian quadrature
zbMATH Keywords: orthogonal polynomial
zbMATH Keywords: weight function
zbMATH Keywords: linear differential operator
describes a project that uses: OPQ
describes a project that uses: HPGT
describes a project that uses: OrthogonalPolynomials
describes a project that uses: Mathematica
MaRDI profile type: Publication
Wikidata QID: Q113878734
full work available at URL: j.cam.2022.114162
OpenAlex ID: W4210852130
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DOI: 10.1016/J.CAM.2022.114162
description: scientific article; zbMATH DE number 7489833