Asymptotic error expansions for hypersingular integrals
type: Item
instance of: scholarly article
title: Asymptotic error expansions for hypersingular integrals
zbMATH Open document ID: 1281.65048
author: Jin Huang
author: Zhu Wang
author: Rui Zhu
published in: Advances in Computational Mathematics
publication date: 2013-04-15T00:00:00Z
review text: The authors consider the hypersingular integral \[ I(g)=f.p.\displaystyle\int_{a}^{b}\frac{g(x)}{|x-t|^{1+\alpha}}dx, \, 0<\alpha\leq 1,\,t\in(a,b), \tag{1} \] where \(f.p.\) denotes the Hadamard finite part of the hypersingular integral. They present quadrature formulae to evaluate (1) for \(0\leq \alpha\leq 1\), which are obtained from the Euler-Maclaurin expansion for the error of the trapezoidal rule. If \(g(x)\) is \(2m\) times differentiable on \([a,b]\), the accuracy order of the algorithms is \(O(h^{2\mu})\) for \(\alpha=1\) and \(O(h^{2\mu-\alpha})\) for \(0<\alpha<1\). Here \(\mu\) is a positive integer determined by the integrand. More accurate results can be achieved, especially, for the hypersingular integration with a periodic integrand. Furthermore, the accuracy order can be greatly improved by applying the extrapolation method. At the same time, this method can be easily extended to evaluate hypersingular integrals with a higher-order singularity. The quadratures only use values of \(g(x)\), but do not depend on its derivatives, which make them easy to use in solving the hypersingular boundary integral equation. Numerical experiments verify the theoretical analysis.
reviewed by: Ana Maria Acu
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 6153902
zbMATH Keywords: hypersingular integral
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: quadrature formula
zbMATH Keywords: Richardson extrapolation
zbMATH Keywords: Hadamard finite part
zbMATH Keywords: Euler-Maclaurin expansion
zbMATH Keywords: algorithm
zbMATH Keywords: numerical experiments
MaRDI profile type: Publication
full work available at URL: s10444-011-9236-x
OpenAlex ID: W2094960234
cites work: Analytical integrations of hypersingular kernel in 3D BEM problems
cites work: Q3996272
cites work: The Numerical Solution of Integral Equations of the Second Kind
cites work: Numerical Quadratures for Singular and Hypersingular Integrals in Boundary Element Methods
cites work: Integral equations with hypersingular kernels--theory and applications to fracture mechanics
cites work: Hypersingular integral equation method for three-dimensional crack problem in shear mode
cites work: Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two-dimensional elasticity
cites work: Improvement of the asymptotic behaviour of the Euler–Maclaurin formula for Cauchy principal value and Hadamard finite‐part integrals
cites work: On the numerical solution of a hypersingular integral equation for elastic scattering from a planar crack
cites work: Q3672341
cites work: Q3323133
cites work: Sigmoidal transformations and the Euler-Maclaurin expansion for evaluating certain Hadamard finite-part integrals
cites work: Regularized product intergration for Hadamard finite part integrals
cites work: Extrapolation Algorithms for Solving Mixed Boundary Integral Equations of the Helmholtz Equation by Mechanical Quadrature Methods
cites work: Evaluations of hypersingular integrals using Gaussian quadrature
cites work: Numerical solution of integral equations with logarithmic-, Cauchy- and Hadamard-type singularities
cites work: On the solution of integral equations with strongly singular kernels
cites work: Linear integral equations
cites work: On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation
cites work: On the numerical solution of a hypersingular integral equation in scattering theory
cites work: Integral equation methods for scattering from an impedance crack.
cites work: Two trigonometric quadrature formulae for evaluating hypersingular integrals
cites work: Numerical quadratures for singular and hypersingular integrals
cites work: Q4831048
cites work: The Splitting Extrapolation Method
cites work: Numerical Quadrature and Asymptotic Expansions
cites work: The Euler-Maclaurin expansion for the Cauchy principal value integral
cites work: Q4764047
cites work: Hypersingular boundary integral equation for axisymmetric elasticity
cites work: The Euler-Maclaurin expansion and finite-part integrals
cites work: A Further Extension of the Euler-Maclaurin Summation Formula
cites work: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals
cites work: Quadrature methods for periodic singular and weakly singular Fredholm integral equations
cites work: Q4284676
cites work: An extrapolation method for a class of boundary integral equations
cites work: Quadratures for boundary integral equations of the first kind with logarithmic kernels
cites work: Multi-parameter extrapolation methods for boundary integral equations
DOI: 10.1007/S10444-011-9236-X
Recommended article: Asymptotic expansions of the error for hyper-singular integrals with an interval variable
Recommended article: Extrapolations for hypersingular integrals
Recommended article: Asymptotic expansions for two-dimensional hypersingular integrals
Recommended article: Asymptotic Error Expansions for Numerical Solutions of Integral Equations
Recommended article: Q4327149
Recommended article: Euler-Maclaurin expansions and approximations of hypersingular integrals
Recommended article: An approximate method for evaluating hypersingular integrals
Recommended article: ON ASYMPTOTIC ERROR EXPANSIONS FOR SINGULAR BOUNDARY VALUE PROBLEMS
Recommended article: An approximate computation method for a class of hypersingular integrals
Recommended article: Q4513087
description: scientific article; zbMATH DE number 6153902