scholarly journals Optimal with respect to accuracy methods for evaluating hypersingular integrals

2021 ◽  
Vol 23 (4) ◽  
pp. 360-378
Author(s):  
Ilya V. Boykov ◽  
Alla I. Boykova
Keyword(s):  
Integral Equations ◽  
Electromagnetic Fields ◽  
Quadrature Formulas ◽  
Power Functions ◽  
Order Of Accuracy ◽  
Hypersingular Integrals ◽  
Hypersingular Integral ◽  
Real Positive Number ◽  
Physical Fields ◽  
Derivatives Of

In this paper we constructed optimal with respect to order quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions Ωur,γ(Ω,M), Ω¯ur,γ(Ω,M), Ω=[−1,1]l, l=1,2,…,M=Const, and γ is a real positive number. The functions that belong to classes Ωur,γ(Ω,M) and Ω¯ur,γ(Ω,M) have bounded derivatives up to the rth order in domain Ω and derivatives up to the sth order (s=r+⌈γ⌉) in domain Ω∖Γ, Γ=∂Ω. Moduli of derivatives of the vth order (r<v≤s) are power functions of d(x,Γ)−1(1+|lnd(x,Γ)|), where d(x,Γ) is a distance between point x and Γ. The interest in these classes of functions is due to the fact that solutions of singular and hypersingular integral equations are their members. Moreover various physical fields, in particular gravitational and electromagnetic fields belong to these classes as well. We give definitions of optimal with respect to accuracy methods for solving hypersingular integrals. We constructed optimal with respect to order of accuracy quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions Ωur,γ(Ω,M) and Ω¯ur,γ(Ω,M).

A Quadrature Formula for a Hypersingular Integral Containing the Weight Function of Jacobi Polynomials with Complex Exponents

2020 ◽  
pp. 94-100
Author(s):  
A.V. Sahakyan
Keyword(s):  
Weight Function ◽  
Integral Equations ◽  
Quadrature Formula ◽  
Jacobi Polynomials ◽  
Quadrature Formulas ◽  
Hölder Continuous ◽  
Approximate Methods ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations ◽  
True Value

Although the concept of a hypersingular integral was introduced by Hadamard at the beginning of the 20th century, it began to be put into practical use only in the second half of the century. The theory of hypersingular integral equations has been widely developed in recent decades and this is due to the fact that they describe the governing equations of many applied problems in various fields: elasticity theory, fracture mechanics, wave diffraction theory, electrodynamics, nuclear physics, geophysics, theory vibrator antennas, aerodynamics, etc. It is analytically possible to calculate the hypersingular integral only for a very narrow class of functions; therefore, approximate methods for calculating such an integral are always in the field of view of researchers and are a rapidly developing area of computational mathematics. There are a very large number of papers devoted to this subject, in which various approaches are proposed both to approximate calculation of the hypersingular integral and to the solution of hypersingular integral equations, mainly taking into account the specifics of the behavior of the densi-ty of the hypersingular integral. In this paper, quadrature formulas are obtained for a hypersingular integral whose density is the product of the Hölder continuous function on the closed interval [–1, 1], and weight function of the Jacobi polynomials . It is assumed that the exponents α and β can be arbitrary complex numbers that satisfy the condition of non-negativity of the real part. The numerical examples show the convergence of the quadrature formula to the true value of the hypersingular integral. The possibility of applying the mechanical quadrature method to the solution of various, including hypersingular, integral equations is indicated.

scholarly journals Computational modeling of hypersingular integral equations for 2D pre-cantor scattering structure

2015 ◽  
Vol 3 (2) ◽  
pp. 161
Author(s):  
Kateryna Nesvit ◽  
Yuriy Gandel
Keyword(s):  
Integral Equations ◽  
Parametric Representation ◽  
Transverse Magnetic ◽  
Quadrature Formulas ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations ◽  
Fractal Properties ◽  
Compact Size ◽  
Diffraction Structure ◽  
Transverse Magnetic Wave

<p>This paper presents the investigative study to derive a computational model based on hypersingular integral equations for the pre-Cantor plane-parallel diffraction structure. Such structure consists of finite numbers of the thin impedance strips located in the XY plane. A plane transverse magnetic wave is incident from infinity on considered diffraction structure at an angle and need to find the total field resulting from the scattering. The model which is considered in this work is an approximation of real fractal antennas in two-dimensional case. Pre-fractal properties of grating allow producing the newest antennas for modern mobile devices due to their compact size and broadband properties. The purpose of this work is to develop computer model their structure using parametric representation of hypersingular integral operator, Nystrom method with specific quadrature formulas. The numerical results have been obtained and investigated for pre-Cantor structures for calculating physics characteristics. These results have been compared and analyzed in different mathematical models and softwares.</p>

3D FRACTURE MECHANICS FOR A MODE I CRACK IN PIEZOELECTRIC MEDIA

2004 ◽  
Vol 01 (03) ◽  
pp. 445-456 ◽  
Author(s):  
MENG-CHENG CHEN
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Electric Displacement ◽  
Basic Density ◽  
Mode I ◽  
Singular Stress ◽  
Hypersingular Integrals ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations ◽  
Electric Displacement Intensity Factor

This paper deals with mode I fracture problems for a planar crack in an infinite piezoelectric solid subjected to electric and tension loading. The finite-part integral concept is used to prove rigidly hypersingular integral equations for the crack by using three-dimensional linear piezoelectricity theory. Investigations on the singularities and the singular stress and electric displacement fields in the vicinity of the crack are made by the dominant-part analysis of the two-dimensional hypersingular integrals. Thereafter the stress and electric displacement intensity factor K-fields and the energy release rate G are exactly obtained by the definitions similar to those of elasticity. Then, a numerical method for the solution of the hypersingular integral equations is developed, in which the displacement and electric potential differences across the crack surfaces are approximated with a product of basic density functions and polynomials. Numerical solutions of several typical planar cracks are obtained with high accuracy.

scholarly journals Integral equations of the first kind of Sonine type

2003 ◽  
Vol 2003 (57) ◽  
pp. 3609-3632 ◽  
Author(s):  
Stefan G. Samko ◽  
Rogério P. Cardoso
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Lebesgue Space ◽  
Inverse Operator ◽  
Orlicz Spaces ◽  
Hypersingular Integral ◽  
The Real ◽  
Integrable Kernel

A Volterra integral equation of the first kindKφ(x):≡∫−∞xk(x−t)φ(t)dt=f(x)with a locally integrable kernelk(x)∈L1loc(ℝ+1)is called Sonine equation if there exists another locally integrable kernelℓ(x)such that∫0xk(x−t)ℓ(t)dt≡1(locally integrable divisors of the unit, with respect to the operation of convolution). The formal inversionφ(x)=(d/dx)∫0xℓ(x−t)f(t)dtis well known, but it does not work, for example, on solutions in the spacesX=Lp(ℝ1)and is not defined on the whole rangeK(X). We develop many properties of Sonine kernels which allow us—in a very general case—to construct the real inverse operator, within the framework of the spacesLp(ℝ1), in Marchaud form:K−1f(x)=ℓ(∞)f(x)+∫0∞ℓ′(t)[f(x−t)−f(x)]dtwith the interpretation of the convergence of this “hypersingular” integral inLp-norm. The description of the rangeK(X)is given; it already requires the language of Orlicz spaces even in the case whenXis the Lebesgue spaceLp(ℝ1).

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