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 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.

Convergence behavior of popular schemes in case of calculating on adaptive grids problems with layers

2020 ◽  
pp. 66-79
Author(s):  
Владимир Дмитриевич Лисейкин ◽  
Виктор Иванович Паасонен
Keyword(s):  
Numerical Experiments ◽  
Order Equation ◽  
Second Order ◽  
Adaptive Grids ◽  
Coordinate Transformations ◽  
Diagonal Dominance ◽  
Counter Flow ◽  
Solution Quality ◽  
Uniform Grids ◽  
New Variables

Проведено сравнение качества решений модельного уравнения второго порядка с малым параметром, полученных по трем различным разностным схемам на специальных адаптивных сетках, явно задаваемых координатным преобразованием, а также на равномерных сетках в новых переменных, соответствующих этому преобразованию. Исследуются схемы второго порядка точности с диагональным преобладанием и без него и простейшая противопотоковая схема. На основе оценок погрешностей сделаны прогнозы относительно свойств решений, подтвержденные анализом и численными экспериментами. Показано, что схема второго порядка аппроксимации с диагональным преобладанием сходится равномерно по малому параметру со вторым порядком лишь в частном случае, когда коэффициент при старшей производной мал только в слое; если же он мал также и вне слоя, порядок сходимости первый. Установлено также, что схема без диагонального преобладания имеет существенно более качественные решения без осцилляций в новых переменных на равномерной сетке, чем в соответствующих им исходных физических координатах. В противоположность ей схемы с диагональным преобладанием не чувствительны к выбору системы координат. The paper compares solution quality to some model second- order equation with a small parameter obtained through three different schemes both on special adaptive grids specified explicitly by coordinate transformations eliminating layers and on uniform grids in a new coordinate related to the transformations. The schemes up to second order in physical and transformation variables both with a diagonal and not diagonal dominance and the simplest counter-flow scheme are analyzed. Predictions of a solution behavior based on estimates of solution errors are described, which are confirmed by numerical experiments and proofs. It is established, in particular, that the scheme of the second order with a diagonal dominance converges uniformly if the coefficient before the second derivative is small at the points of the boundary layer only. It was also demonstrated for the schemes without a diagonal dominance, mach better solutions without oscillations are obtained on uniform grids in new variables than on corresponding adaptive grids in the original physical coordinates.

scholarly journals A High-Order Numerical Method for a Nonlinear System of Second-Order Boundary Value Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Minqiang Xu ◽  
Jing Niu ◽  
Li Guo
Keyword(s):  
Nonlinear System ◽  
Boundary Value Problems ◽  
Numerical Experiments ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Boundary Value ◽  
Linear Equations ◽  
Second Order ◽  
High Order ◽  
Reproducing Kernel Method

This paper is concerned with a high-order numerical scheme for nonlinear systems of second-order boundary value problems (BVPs). First, by utilizing quasi-Newton’s method (QNM), the nonlinear system can be transformed into linear ones. Based on the standard Lobatto orthogonal polynomials, we introduce a high-order Lobatto reproducing kernel method (LRKM) to solve these linear equations. Numerical experiments are performed to investigate the reliability and efficiency of the presented method.

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