scholarly journals Solution of Cauchy-Type Singular Integral Equations of the First Kind with Zeros of Jacobi Polynomials as Interpolation Nodes

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
G. E. Okecha
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Jacobi Polynomials ◽  
Singular Integral Equations ◽  
Quadrature Rule ◽  
Convergence Result ◽  
Cauchy Type ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Lagrangian Interpolation

Of concern in this paper is the numerical solution of Cauchy-type singular integral equations of the first kind at a discrete set of points. A quadrature rule based on Lagrangian interpolation, with the zeros of Jacobi polynomials as nodes, is developed to solve these equations. The problem is reduced to a system of linear algebraic equations. A theoretical convergence result for the approximation is provided. A few numerical results are given to illustrate and validate the power of the method developed. Our method is more accurate than some earlier methods developed to tackle this problem.

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 Finite-part singular integral approximations in Hilbert spaces

2004 ◽  
Vol 2004 (52) ◽  
pp. 2787-2793
Author(s):  
E. G. Ladopoulos ◽  
G. Tsamasphyros ◽  
V. A. Zisis
Keyword(s):  
Unit Circle ◽  
Integral Equations ◽  
Singular Integral ◽  
Hilbert Spaces ◽  
Existence Theorems ◽  
Singular Integral Equations ◽  
Finite Part ◽  
Algebraic Equations ◽  
Integration Interval ◽  
Linear Algebraic Equations

Some new approximation methods are proposed for the numerical evaluation of the finite-part singular integral equations defined on Hilbert spaces when their singularity consists of a homeomorphism of the integration interval, which is a unit circle, on itself. Therefore, some existence theorems are proved for the solutions of the finite-part singular integral equations, approximated by several systems of linear algebraic equations. The method is further extended for the proof of the existence of solutions for systems of finite-part singular integral equations defined on Hilbert spaces, when their singularity consists of a system of diffeomorphisms of the integration interval, which is a unit circle, on itself.

scholarly journals Singular Integral Equations of Convolution Type with Cosecant Kernels and Periodic Coefficients

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Pingrun Li
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Complex Analysis ◽  
Boundary Value ◽  
Singular Integral Equations ◽  
Convolution Type ◽  
Algebraic Equations ◽  
Periodic Coefficients ◽  
Linear Algebraic Equations ◽  
Classical Boundary

We study singular integral equations of convolution type with cosecant kernels and periodic coefficients in class L2[-π,π]. Such equations are transformed into a discrete jump problem or a discrete system of linear algebraic equations by using discrete Fourier transform. The conditions of Noethericity and the explicit solutions are obtained by means of the theory of classical boundary value problem and of the Fourier analysis theory. This paper will be of great significance for the study of improving and developing complex analysis, integral equations, and boundary value problems.

Plane Analysis of Finite Multilayered Media With Multiple Aligned Cracks—Part I: Theory

2006 ◽  
Vol 74 (1) ◽  
pp. 128-143 ◽  
Author(s):  
Linfeng Chen ◽  
Marek-Jerzy Pindera
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Mixed Boundary ◽  
Plane Deformation ◽  
Singular Integral Equations ◽  
Cauchy Type ◽  
Algebraic Equations ◽  
Energy Release Rates ◽  
Global Stiffness Matrix ◽  
Bounding Surfaces

Elasticity solutions are developed for finite multilayered domains weakened by aligned cracks that are in a state of generalized plane deformation under two types of end constraints. Multilayered domains consist of an arbitrary number of finite-length and finite-height isotropic, orthotropic or monoclinic layers typical of differently oriented, unidirectionally reinforced laminas arranged in any sequence in the plane in which the analysis is conducted. The solution methodology admits any number of arbitrarily distributed interacting or noninteracting cracks parallel to the horizontal bounding surfaces at specified elevations or interfaces. Based on half-range Fourier series and the local/global stiffness matrix approach, the mixed boundary-value problem is reduced to a system of coupled singular integral equations of the Cauchy type with kernels formulated in terms of the unknown displacement discontinuities. Solutions to these integral equations are obtained by representing the unknown interfacial displacement discontinuities in terms of Jacobi or Chebyshev polynomials with unknown coefficients. The application of orthogonality properties of these polynomials produces a system of algebraic equations that determines the unknown coefficients. Stress intensity factors and energy release rates are derived from dominant parts of the singular integral equations. In Part I of this paper we outline the analytical development of this technique. In Part II we verify this solution and present new fundamental results relevant to the existing and emerging technologies.

scholarly journals Solvability of singular integral equations with rotations and degenerate kernels in the vanishing coefficient case

2014 ◽  
Vol 13 (01) ◽  
pp. 1-21 ◽  
Author(s):  
L. P. Castro ◽  
E. M. Rojas ◽  
S. Saitoh ◽  
N. M. Tuan ◽  
P. D. Tuan
Keyword(s):  
Boundary Value Problems ◽  
Unit Circle ◽  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Degenerate Kernel ◽  
Algebraic Equations ◽  
Explicit Formulas ◽  
Riemann Boundary ◽  
Linear Algebraic Equations

By means of Riemann boundary value problems and of certain convenient systems of linear algebraic equations, this paper deals with the solvability of a class of singular integral equations with rotations and degenerate kernel within the case of a coefficient vanishing on the unit circle. All the possibilities about the index of the coefficients in the corresponding equations are considered and described in detail, and explicit formulas for their solutions are obtained. An example of application of the method is shown at the end of the last section.

On the receding contact between a graded and a homogeneous layer due to a flat-ended indenter

2021 ◽  
pp. 108128652110431
Author(s):  
Rui Cao ◽  
Changwen Mi
Keyword(s):  
Finite Element ◽  
Contact Problem ◽  
Integral Equations ◽  
Singular Integral ◽  
Functional Dependence ◽  
Singular Integral Equations ◽  
Homogeneous Layer ◽  
Cauchy Type ◽  
Receding Contact ◽  
Contact Pressures

This paper solves the frictionless receding contact problem between a graded and a homogeneous elastic layer due to a flat-ended rigid indenter. Although its Poisson’s ratio is kept as a constant, the shear modulus in the graded layer is assumed to exponentially vary along the thickness direction. The primary goal of this study is to investigate the functional dependence of both contact pressures and the extent of receding contact on the mechanical and geometric properties. For verification and validation purposes, both theoretical analysis and finite element modelings are conducted. In the analytical formulation, governing equations and boundary conditions of the double contact problem are converted into dual singular integral equations of Cauchy type with the help of Fourier integral transforms. In view of the drastically different singularity behavior of the stationary and receding contact pressures, Gauss–Chebyshev quadratures and collocations of both the first and the second kinds have to be jointly used to transform the dual singular integral equations into an algebraic system. As the resultant algebraic equations are nonlinear with respect to the extent of receding contact, an iterative algorithm based on the method of steepest descent is further developed. The semianalytical results are extensively verified and validated with those obtained from the graded finite element method, whose implementation details are also given for easy reference. Results from both approaches reveal that the property gradation, indenter width, and thickness ratio all play significant roles in the determination of both contact pressures and the receding contact extent. An appropriate combination of these parameters is able to tailor the double contact properties as desired.

Interaction of Three Interfacial Cracks between an Orthotropic Half-Plane Bonded to a Dissimilar Orthotropic Layer with Punch

2017 ◽  
Vol 72 (11) ◽  
pp. 1021-1029
Author(s):  
P.K. Mishra ◽  
P. Singh ◽  
S. Das
Keyword(s):  
Integral Equations ◽  
Half Plane ◽  
Singular Integral ◽  
Jacobi Polynomials ◽  
Singular Integral Equations ◽  
Orthotropic Layer ◽  
Central Crack ◽  
Interfacial Cracks ◽  
Crack Shielding

AbstractThis article deals with the interactions between a central crack and a pair of outer cracks situated at the interface of an orthotropic elastic half-plane bonded to a dissimilar orthotropic layer with a punch. The problem is reduced to the solution of three simultaneous singular integral equations that are finally solved using Jacobi polynomials. The phenomena of crack shielding and crack amplification have been depicted through graphs for different particular cases.

A Boundary Element Method for Plane Anisotropic Elastic Media

1990 ◽  
Vol 57 (3) ◽  
pp. 600-606 ◽  
Author(s):  
Kyu J. Lee ◽  
A. K. Mal
Keyword(s):  
Integral Equations ◽  
Complex Plane ◽  
Singular Integral Equations ◽  
Boundary Curve ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Numerical Formulation ◽  
Complex Linear ◽  
Anisotropic Elastic ◽  
Improved Accuracy

The general problem of plane anisotropic elastostatics is formulated in terms of a system of singular integral equations with Cauchy kernels by means of the classical stress function approach. The integral equations are represented over the image of the boundary in the complex plane and a numerical scheme is developed for their solution. The boundary curve is discretized and suitable polynomial approximations of the unknown functions in terms of the complex variable are introduced. This reduces the equations to a set of complex linear algebraic equations which can be inverted to yield the stresses in a straightforward manner. The major difference between the present technique and the previous ones is in the numerical formulation. The integral equations are discretized in the complex plane and not in terms of real variables which depend on arc length, resulting in improved accuracy in presence of strong boundary curvature.

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