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.

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.

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 On the solution of one integro-differential equation with singular and hypersingular integrals

2020 ◽  
Vol 56 (3) ◽  
pp. 298-309
Author(s):  
A. P. Shilin
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Complex Plane ◽  
Continuation Method ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Hypersingular Integrals ◽  
Integro Differential Equation ◽  
Linear Differential ◽  
Special Points

A linear integro-differential equation of the first order given on a closed curve located on the complex plane is studied. The coefficients of the equation have a special structure. The equation contains a singular integral, which can be understood as the main value by Cauchy, and a hypersingular integral which can be understood as the end part by Hadamard. The analytical continuation method is applied. The equation is reduced to a sequential solution of the Riemann boundary value problem and two linear differential equations. The Riemann problem is solved in the class of analytic functions with special points. Differential equations are solved in the class of analytical functions on the complex plane. The conditions for the solvability of the original equation are explicitly given. The solution of the equation when these conditions are fulfilled is also given explicitly. Examples are considered. A non-obvious special case is analyzed.

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.

Multi-parameter Singular Integrals. (AM-189)

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Partial Differential Equations ◽  
Graduate Students ◽  
General Theory ◽  
Differential Equations ◽  
Singular Integrals ◽  
Main Tool ◽  
Classical Theorem ◽  
Quantitative Version ◽  
Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

Sign in / Sign up

Export Citation Format

Share Document