Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

Approximate Solution of a Singular Integral Equation with a Cauchy Kernel on the Euclidean Plane

2018 ◽  
Vol 18 (4) ◽  
pp. 741-752
Author(s):  
Dorota Pylak ◽  
Paweł Karczmarek ◽  
Paweł Wójcik
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Euclidean Plane ◽  
Singular Integral Equations ◽  
Cauchy Kernel ◽  
Challenging Problem ◽  
One Dimensional ◽  
Interpolating Polynomials

AbstractMultidimensional singular integral equations (SIEs) play a key role in many areas of applied science such as aerodynamics, fluid mechanics, etc. Solving an equation with a singular kernel can be a challenging problem. Therefore, a plethora of methods have been proposed in the theory so far. However, many of them are discussed in the simplest cases of one–dimensional equations defined on the finite intervals. In this study, a very efficient method based on trigonometric interpolating polynomials is proposed to derive an approximate solution of a SIE with a multiplicative Cauchy kernel defined on the Euclidean plane. Moreover, an estimation of the error of the approximated solution is presented and proved. This assessment and an illustrating example show the effectiveness of our proposal.

scholarly journals Application of Jacobi Polynomials to the Approximate Solution of a Singular Integral Equation with Cauchy Kernel on the Real Half-line

2006 ◽  
Vol 6 (3) ◽  
pp. 326-335
Author(s):  
D. Pylak
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Jacobi Polynomials ◽  
Approximate Solutions ◽  
Half Line ◽  
Cauchy Kernel ◽  
The Real

AbstractIn this paper, exact solution of the characteristic equation with Cauchy kernel on the real half-line is presented. Next, Jacobi polynomials are used to derive approximate solutions of this equation. Moreover, estimations of errors of the approximated solutions are presented and proved.

scholarly journals A Galerkin-Petrov method for singular integral equations

1983 ◽  
Vol 25 (2) ◽  
pp. 261-275 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Convergence Theorem ◽  
Singular Integral Equations ◽  
Approximate Solutions ◽  
Principal Result ◽  
Cauchy Kernel

AbstractA Galerkin-Petrov method for the approximate solution of the complete singular integral equation with Cauchy kernel, based upon the use of two sets of orthogonal polynomials, is considered. The principal result of this paper proves convergence of the approximate solutions to the exact solution making use of a convergence theorem previously given by the author. In conclusion, some related topics such as a first iterate of the approximate solution and a discretized Galerkin-Petrov method are considered. The paper extends to a much more general equation many results obtained by other authors in particular cases.

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 Torsional Impact Response of a Penny-Shaped Interface Crack in Bonded Materials With a Graded Material Interlayer

2002 ◽  
Vol 69 (3) ◽  
pp. 303-308 ◽  
Author(s):  
C. Li ◽  
Z. Duan ◽  
Z. Zou
Keyword(s):  
Integral Equation ◽  
Stress Intensity ◽  
Singular Integral Equation ◽  
Interface Crack ◽  
Singular Integral ◽  
Dynamic Stress ◽  
Mixed Boundary ◽  
Dynamic Stress Intensity Factor ◽  
Cauchy Kernel ◽  
Dynamic Stress Intensity Factors

In this paper, the dynamic response of a penny-shaped interface crack in bonded dissimilar homogeneous half-spaces is studied. It is assumed that the two materials are bonded together with such a inhomogeneous interlayer that makes the elastic modulus in the direction perpendicular to the crack surface is continuous throughout the space. The crack surfaces are assumed to be subjected to torsional impact loading. Laplace and Hankel integral transforms are applied combining with a dislocation density function to reduce the mixed boundary value problem into a singular integral equation with a generalized Cauchy kernel in Laplace domain. By solving the singular integral equation numerically and using a numerical Laplace inversion technique, the dynamic stress intensity factors are obtained. The influences of material properties and interlayer thickness on the dynamic stress intensity factor are investigated.

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