Solution of a Linear Integral Equation of Third Kind

2002 ◽  
Vol 9 (1) ◽  
pp. 179-196
Author(s):  
D. Shulaia
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Fredholm Theory ◽  
Linear Integral Equation ◽  
Hölder Functions ◽  
Linear Integral ◽  
Necessary And Sufficient

Abstract The aim of this paper is to study, in the class of Hölder functions, a nonhomogeneous linear integral equation with coefficient cos 𝑥. Necessary and sufficient conditions for the solvability of this equation are given under some assumptions on its kernel. The solution is constructed analytically, using the Fredholm theory and the theory of singular integral equations.

scholarly journals A convergence theorem for singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 539-552 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Convergence Theory ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

scholarly journals Toeplitz operators and Wiener-Hopf factorisation: an introduction

2017 ◽  
Vol 4 (1) ◽  
pp. 130-145 ◽  
Author(s):  
M. Cristina Câmara
Keyword(s):  
Integral Equations ◽  
Half Plane ◽  
Singular Integral ◽  
Hardy Spaces ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Necessary And Sufficient Conditions ◽  
Upper Half Plane ◽  
Necessary And Sufficient

Abstract Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces Hp of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in L−1(ℝ)

scholarly journals The properties of solutions of weakly singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 419-430 ◽  
Author(s):  
G. Vainikko ◽  
A. Pedas
Keyword(s):  
Integral Equation ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Linear Integral Equation ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations ◽  
The Difference ◽  
Linear Integral ◽  
Differential Properties ◽  
Derivatives Of

AbstractWe examine the differential properties of the solution of the linear integral equation of the second kind, whose kernel depends on the difference of arguments and has an integrable singularity at the point zero. The derivatives of the solution of the equation have singularities at the end points of the domain of integration, and we derive precise estimates for these singularities.

scholarly journals On the approximate solution of nonlinear singular integral equations with positive index

1996 ◽  
Vol 19 (2) ◽  
pp. 389-396 ◽  
Author(s):  
S. M. Amer
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Collocation Method ◽  
Jordan Curve ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Holder Space ◽  
Smooth Jordan Curve ◽  
Positive Index

This paper is devoted to investigating a class of nonlinear singular integral equations with a positive index on a simple closed smooth Jordan curve by the collocation method. Sufficient conditions are given for the convergence of this method in Holder space.

Boundedness of weighted singular integral operators in grand Lebesgue spaces

2011 ◽  
Vol 18 (2) ◽  
pp. 259-269
Author(s):  
Vakhtang Kokilashvili ◽  
Stefan Samko
Keyword(s):  
Singular Integral ◽  
Lebesgue Space ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Lebesgue Spaces ◽  
Grand Lebesgue Spaces ◽  
Carleson Curve ◽  
Power Weights ◽  
Necessary And Sufficient ◽  
Grand Lebesgue Space

Abstract We obtain the necessary and sufficient conditions for the boundedness of the weighted singular integral operator with power weights in grand Lebesgue spaces. Because of applications to singular integral equations, the underlying set on which the functions are defined is a Carleson curve in the complex plane. Note that weighted boundedness of an operator in grand Lebesgue space is not the same as the boundedness in weighted grand Lebesgue space.

Regularization Method for Complete Singular Integral Equation with Hilbert Kernel on Open Arcs

2013 ◽  
Vol 765-767 ◽  
pp. 643-646
Author(s):  
Li Xia Cao
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Regularization Method ◽  
Singular Integral Equations ◽  
Noether Theorem ◽  
Hilbert Kernel

We considered the regularization method for a kind of complete singular integral equation with Hilbert kernel on open arcs lying in a period strip. And based on this, we obtained the solvable Noether theorem for this kind of complete singular integral equations.

scholarly journals Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 872 ◽  
Author(s):  
◽  
Shuhuang Xiang ◽  
Guidong Liu
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Numerical Solution ◽  
Integral Equations ◽  
Kernel Function ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Cauchy Type ◽  
Error Analyses ◽  
Highly Oscillatory

This paper aims to present a Clenshaw–Curtis–Filon quadrature to approximate thesolution of various cases of Cauchy-type singular integral equations (CSIEs) of the second kind witha highly oscillatory kernel function. We adduce that the zero case oscillation (k = 0) proposed methodgives more accurate results than the scheme introduced in Dezhbord at el. (2016) and Eshkuvatovat el. (2009) for small values of N. Finally, this paper illustrates some error analyses and numericalresults for CSIEs.

On an approximate solution of a class of surface singular integral equations of the first kind

2020 ◽  
Vol 27 (1) ◽  
pp. 97-102 ◽  
Author(s):  
Elnur H. Khalilov
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Neumann Boundary ◽  
Neumann Boundary Value Problems

AbstractIn this work, a method for calculating an approximate solution of a singular integral equation of first kind is presented for the Neumann boundary value problems for the Helmholtz equation.

scholarly journals The method for calculating singular integrals in problems of axially symmetric Stokes flows

2019 ◽  
Keyword(s):  
Integral Equation ◽  
Integral Representation ◽  
Integral Equations ◽  
Singular Integral ◽  
Geometric Mean ◽  
Singular Integral Equations ◽  
Elliptic Integrals ◽  
Quadrature Formulas ◽  
Axially Symmetric ◽  
One Dimensional

The flow of a viscous fluid at small Reynolds numbers (Stokes flow) in a three-dimensional formulation is investigated. In this case, the inertial terms in the equations of motion can be neglected. Such flows can occur in nanotubes that can be considered as inclusions in representative volume elements of nanomaterials. By using the fundamental solution of Ossen, an integral representation of the velocity is proposed. This representation is used to receive an integral equation for an unknown density. The solution of the resulting equation makes it possible to calculate the fluid pressure on the walls of the shell. The case of axially symmetric flows is investigated. For this, an integral representation of the unknown velocity in cylindrical coordinates is obtained. By integrating over the circumferential coordinate, the two-dimensional singular integral equation is reduced to one-dimensional one. It has been proved that the components of the kernels in singular operators are expressed in terms of elliptic integrals of the first and second kind. It has been proved that the singularities of the kernels of one-dimensional singular integral equations have a logarithmic character. To calculate elliptic integrals, the Gaussian algorithm based on the use of the arithmetic-geometric mean value is proposed. This procedure allows us to obtain logarithmic singular components with high accuracy, which makes it possible to use special quadrature formulas to calculate such integrals. An algorithm with usage of the boundary element method for the numerical solution of the obtained singular integral equations is proposed. The method for solving one-dimensional singular equations, where the kernels contain elliptic integrals with logarithmic singularities (i.e logarithmic singularity is not expressed explicitly) has been tested. The obtained numerical results have been compared with the well-known analytical solutions. The data obtained indicate the high efficiency of the proposed numerical method.

scholarly journals Equilibrium internal fractures in elastic bodies supported by thin flexible coatings

2020 ◽  
Vol 47 (3) ◽  
pp. 111-121
Author(s):  
B. V. Sobol ◽  
E. V. Rashidova ◽  
P. V. Vasiliev ◽  
A. I. Novikova
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Influence Factor ◽  
Singular Integral Equations ◽  
The Body ◽  
Elastic Bodies ◽  
Flexible Material ◽  
Flexible Coating

Objective. In this paper, the authors study problems of a plane strain of elastic bodies containing internal rectilinear fractures. In each case, the margins of the considered areas are supported by thin flexible coatings. The first part of the paper is devoted to the problem of an infinite elastic wedge, the faces of which are free from the outside and reinforced with a thin flexible material, and the bisector contains a rectilinear fracture with regular forces applied to the margins, and to the study of the stress concentration at the fracture vertices. In the second part of the paper, the authors consider the problem of an equilibrium radial internal fracture in the cross-section of a round pipe. The inner surface of the pipe experiences hydrostatic pressure; the outer surface is reinforced with a thin flexible coating. The purpose of the study in each of the presented tasks is to determine the values of the influence factor. Methods. Both problems are united by a single approach, in which the presence of a coating is modeled mathematically, using special marginal conditions obtained based on an asymptotic analysis of the exact solution for a strip or ring flexible coating of small relative thickness. In the first issue, the singular integral equation is derived using the Mellin transform, which allows proceeding to the solution of a system of ordinary differential equations and obtaining a singular integral equation relative to the derivative of the discontinuity function of the first kind with a Cauchy kernel. In the second issue, discontinuous solutions are constructed using the Fourier series, resulting in a singular integral equation of a similar structure. Previously, similar ideas were successfully implemented by the authors in the study of the problem of the equilibrium state of a strip with a coating weakened by an internal transverse fracture under arbitrary conditions on the lower edge of the strip. Conclusion. Singular integral equations for the considered problems are obtained. The collocation method is used to construct solutions of singular integral equations for various combinations of geometric and physical characteristics of issues. In all the considered cases, the values of the influence factor were calculated. The analysis of changes in the influence factor depending on various combinations of geometric parameters and mechanical characteristics of problems is carried out. It is noted that with increasing rigidity of the coating and increasing its thickness, the values of the influence factor decrease; the increase in the value of the influence factor is provided by approaching the fracture to the body margin and increasing its relative length.

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