Compact Numerical Quadrature Formulas for Hypersingular Integrals and Integral Equations

2012 ◽  
Vol 54 (1) ◽  
pp. 145-176 ◽  
Author(s):  
Avram Sidi
Keyword(s):  
Integral Equations ◽  
Numerical Quadrature ◽  
Quadrature Formulas ◽  
Hypersingular Integrals

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.

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