Quadrature rules for weakly singular, strongly singular, and hypersingular integrals in boundary integral equation methods

Tsalamengas, J.L.

doi:10.1016/j.jcp.2015.09.053

Cited by 23 publications

(16 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…For the sake of completeness, it should be also mentioned that convergent numerical solutions to electromagnetics IEs with smooth and singular kernels can be obtained using the Nystrom‐type algorithms developed by Tsalamengas [, , ], Balaban et al [], Bulygin et al [], and others. Such algorithms do not use partial inversion and, instead, rely on the theorems of approximation of integrals with quadratures.…”

Section: Introductionmentioning

confidence: 99%

Method of analytical regularization in computational photonics

Nosich¹

2016

Radio Science

View full text Add to dashboard Cite

We discuss the advantages of the conversion of electromagnetic field problems to the Fredholm second‐kind integral equations (analytical regularization) and Fredholm second‐kind infinite‐matrix equations (analytical preconditioning). Special attention is paid to specific features of the characterization of metals and dielectrics in the optical range and their effect on the problem formulation and on the methods applicable to the mentioned conversion.

show abstract

Section: Introductionmentioning

confidence: 99%

Method of analytical regularization in computational photonics

Nosich¹

2016

Radio Science

View full text Add to dashboard Cite

show abstract

“…Bernstein polynomials have been employed to obtain the approximate solution of Abel's integral equation . An efficient approach based on n ‐point Gauss‐Gegenbauer quadrature rules for weakly singular, strongly singular, and hypersingular integrals that arise in integral equation formulations of potential problems has been applied in Tsalamengas . The Adomian decomposition method and the variational iteration method have been used to provide the approximate solution of the Volterra integral equation with a weakly singular kernel in the reproducing kernel space .…”

Section: Introductionmentioning

confidence: 99%

“…7 An efficient approach based on n-point Gauss-Gegenbauer quadrature rules for weakly singular, strongly singular, and hypersingular integrals that arise in integral equation formulations of potential problems has been applied in Tsalamengas. 8 The Adomian decomposition method and the variational iteration method have been used to provide the approximate solution of the Volterra integral equation with a weakly singular kernel in the reproducing kernel space. 9 Collocation method based on the Bernstein polynomials has been introduced for the approximate solution of a class of linear Volterra integro-differential equations with weakly singular kernel in Işik et al 10 Also, there are few numerical methods for solving partial integro-differential equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical scheme for solving singular fractional partial integro‐differential equation via orthonormal Bernoulli polynomials

Samadyar

Mirzaee

2019

Int J Numerical Modelling

View full text Add to dashboard Cite

In this paper, an efficient matrix method based on 2D orthonormal Bernoulli polynomials are developed to obtain numerical solution of weakly singular fractional partial integro‐differential equations (FPIDEs). Operational matrix of integration, almost operational matrix of integration, and product operational matrix are employed to transform the solution of Volterra singular FPIDEs to the system of linear or nonlinear algebraic equations. The obtained algebraic system can be easily solved by using an appropriate numerical method. Also, some useful theorems concerning the convergence and error analysis associated to the mentioned approach are proven in the current paper. Finally, some numerical examples are included to illustrate the accuracy, efficiency, and applicability of the proposed approach.

show abstract

“…One fact is that a great variety of methods to evaluate integrals with strong singularities are currently known, most of which have been published after the sixth decade of the 20th century (cf. [5,7,8]). When ε is small, the integrals on the right side of (2) are nearly hypersingular, an issue also treated by several authors due to the important role they play in the applications of the BEM (cf.…”

Section: Introductionmentioning

confidence: 99%

Evaluation of finite part integrals using a regularization technique that decreases instability

Illn-Gonzlez

Rebollido-Lorenzo

2017

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

A hypersingular integral can be regularized by replacing the whole integrand by a forward difference quotient of 2nd order. If the density function is nearly singular, then Gauss quadrature formulas associated with a suitable modification of the Chebyshev weight function allow to obtain great precision with few nodes. However, in most cases, the own nature of this procedure makes unpredictable the location of quadrature nodes. This paper presents a simple but effective technique whose aim is to mitigate instability when some node lies too close to the pole. Some numerical examples are shown to evaluate the performance of the proposed method.

show abstract

Quadrature rules for weakly singular, strongly singular, and hypersingular integrals in boundary integral equation methods

Cited by 23 publications

References 31 publications

Method of analytical regularization in computational photonics

Method of analytical regularization in computational photonics

Numerical scheme for solving singular fractional partial integro‐differential equation via orthonormal Bernoulli polynomials

Evaluation of finite part integrals using a regularization technique that decreases instability

Contact Info

Product

Resources

About