scholarly journals Error bounds for a Gauss-type quadrature rule to evaluate hypersingular integrals

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2525-2543
Author(s):  
Bonis de ◽  
Donatella Occorsio
Keyword(s):  
Quadrature Rule ◽  
Density Functions ◽  
Finite Part ◽  
Type Condition ◽  
Hypersingular Integrals ◽  
Numerical Tests ◽  
Hadamard Finite Part ◽  
Gauss Type ◽  
Computational Aspects ◽  
The Stability

In the present paper we consider hypersingular integrals of the following type =?+?,0 f(x)/(x-t)p+1 w?(x)dx, where the integral is understood in the Hadamard finite part sense, p is a positive integer, w?(x) = e-xx? is a Laguerre weight of parameter ? ? 0 and t > 0. In [6] we proposed an efficient numerical algorithm for approximating (1), focusing our attention on the computational aspects and on the efficient implementation of the method. Here, we introduce the method discussing the theoretical aspects, by proving the stability and the convergence of the procedure for density functions f s.t. f(p) satisfies a Dini-type condition. For the sake of completeness, we present some numerical tests which support the theoretical estimates.

AUTOMATIC QUADRATURE SCHEME FOR EVALUATING HYPERSINGULAR INTEGRALS

2012 ◽  
Vol 09 ◽  
pp. 581-585
Author(s):  
SUZAN J. OBAIYS ◽  
Z. K. ESKHUVATOV ◽  
N. M. A. NIK LONG
Keyword(s):  
Numerical Experiments ◽  
Second Order ◽  
Close Connection ◽  
Finite Part ◽  
Smooth Functions ◽  
Hypersingular Integrals ◽  
Quadrature Scheme ◽  
Hadamard Finite Part ◽  
Automatic Quadrature

Hasegawa constructed the automatic quadrature scheme (AQS), of Cauchy principle value integrals for smooth functions. There is a close connection between Hadamard and Cauchy principle value integral. In this paper, we modify AQS for hypersingular integrals with second-order singularities, using hasegawa's formula and based on the relations between Hadamard finite part integral and Cauchy principle value integral. Numerical experiments are also given, to validate the modified AQS.

scholarly journals Integro-differential equations with singular and hypersingular integrals

2019 ◽  
Vol 54 (4) ◽  
pp. 404-407 ◽  
Author(s):  
E. I. Zverovich ◽  
A. P. Shilin
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Singular Integrals ◽  
Closed Curve ◽  
Special Structure ◽  
Finite Part ◽  
Hypersingular Integrals ◽  
Hadamard Finite Part

Quadrature linear integro-differential equations on a closed curve located in the complex plane are solved. The equations contain singular integrals which are understood in the sense of the main value and hypersingular integrals which are understood in the sense of the Hadamard finite part. The coefficients of the equations have a special structure.

Hypersingular Boundary Integral Equations: Some Applications in Acoustic and Elastic Wave Scattering

1990 ◽  
Vol 57 (2) ◽  
pp. 404-414 ◽  
Author(s):  
G. Krishnasamy ◽  
L. W. Schmerr ◽  
T. J. Rudolphi ◽  
F. J. Rizzo
Keyword(s):  
Boundary Integral Equations ◽  
Acoustic Scattering ◽  
Boundary Integral ◽  
Three Dimensions ◽  
Finite Part ◽  
Important Special Case ◽  
Hypersingular Integrals ◽  
Hadamard Finite Part ◽  
Spatial Dimensions ◽  
Regular Line

The properties of hypersingular integrals, which arise when the gradient of conventional boundary integrals is taken, are discussed. Interpretation in terms of Hadamard finite-part integrals, even for integrals in three dimensions, is given, and this concept is compared with the Cauchy Principal Value, which, by itself, is insufficient to render meaning to the hypersingular integrals. It is shown that the finite-part integrals may be avoided, if desired, by conversion to regular line and surface integrals through a novel use of Stokes’ theorem. Motivation for this work is given in the context of scattering of time-harmonic waves by cracks. Static crack analysis of linear elastic fracture mechanics is included as an important special case in the zero-frequency limit. A numerical example is given for the problem of acoustic scattering by a rigid screen in three spatial dimensions.

scholarly journals Numerical treatment of the generalized Love integral equation

2020 ◽  
Author(s):  
Luisa Fermo ◽  
Maria Grazia Russo ◽  
Giada Serafini
Keyword(s):  
Integral Equation ◽  
Quadrature Formula ◽  
Numerical Procedure ◽  
Quadrature Rule ◽  
Weighted Spaces ◽  
Stability And Convergence ◽  
Numerical Treatment ◽  
Nyström Method ◽  
Numerical Tests ◽  
The Stability

Abstract In this paper, the generalized Love integral equation has been considered. In order to approximate the solution, a Nyström method based on a mixed quadrature rule has been proposed. Such a rule is a combination of a product and a “dilation” quadrature formula. The stability and convergence of the described numerical procedure have been discussed in suitable weighted spaces and the efficiency of the method is shown by some numerical tests.

Fast and high-precision calculation of earth return mutual impedance between conductors over a multilayered soil

2018 ◽  
Vol 37 (3) ◽  
pp. 1214-1227 ◽  
Author(s):  
Junjie Ma
Keyword(s):  
Quadrature Rule ◽  
Content Type ◽  
Mutual Impedance ◽  
Numerical Tests ◽  
Sommerfeld Integral ◽  
The Earth ◽  
Finite Integral ◽  
Infinite Integral ◽  
Improper Integrals ◽  
Differential Matrix

Purpose Solutions for the earth return mutual impedance play an important role in analyzing couplings of multi-conductor systems. Generally, the mutual impedance is approximated by Pollaczek integrals. The purpose of this paper is devising fast algorithms for calculation of this kind of improper integrals and its applications. Design/methodology/approach According to singular points, the Pollaczek integral is divided into two parts: the finite integral and the infinite integral. The finite part is computed by combining an efficient Levin method, which is implemented with a Chebyshev differential matrix. By transforming the integration path, the tail integral is calculated with help of a transformed Clenshaw–Curtis quadrature rule. Findings Numerical tests show that this new method is robust to high oscillation and nearly singularities. Thus, it is suitable for evaluating Pollaczek integrals. Furthermore, compared with existing method, the presented algorithm gives high-order approaches for the earth return mutual impedance between conductors over a multilayered soil with wide ranges of parameters. Originality/value An efficient truncation strategy is proposed to accelerate numerical calculation of Pollaczek integral. Compared with existing algorithms, this method is easier to be applied to computation of similar improper integrals, such as Sommerfeld integral.

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