Hypersingular integral equations on curved boundaries

1995 ◽  
pp. 2788-2793
Author(s):  
Roland Maucher
Keyword(s):  
Integral Equations ◽  
Curved Boundaries ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations

scholarly journals Hypersingular integral equations of the first kind: A modified homotopy perturbation method and its application to vibration and active control

2019 ◽  
Vol 38 (2) ◽  
pp. 706-727 ◽  
Author(s):  
Reza Novin ◽  
Mohammad Ali Fariborzi Araghi
Keyword(s):  
Perturbation Method ◽  
Integral Equations ◽  
Homotopy Perturbation Method ◽  
The Novel ◽  
Validity And Reliability ◽  
Hypersingular Integral ◽  
Homotopy Perturbation ◽  
Hypersingular Integral Equations ◽  
Ideal Tool ◽  
Novel Method

This paper attempts to propose and investigate a modification of the homotopy perturbation method to study hypersingular integral equations of the first kind. Along with considering this matter, of course, the novel method has been compared with the standard homotopy perturbation method. This method can be conveniently fast to get the exact solutions. The validity and reliability of the proposed scheme are discussed. Different examples are included to prove so. According to the results, we further state that new simple homotopy perturbation method is so efficient and promises the exact solution. The modification of the homotopy perturbation method has been discovered to be the significant ideal tool in dealing with the complicated function-theoretic analytical structures within an analytical method.

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.

Sign in / Sign up

Export Citation Format

Share Document