A Galerkin-Petrov method for singular integral equations

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.

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

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2006-0018 ◽

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.

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

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2017-0052 ◽

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.

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

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2008-0010 ◽

2008 ◽

Vol 8 (2) ◽

pp. 143-154 ◽

Cited By ~ 1

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.

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A convergence theorem for singular integral equations

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s033427000000285x ◽

1981 ◽

Vol 22 (4) ◽

pp. 539-552 ◽

Cited By ~ 10

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.

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Application of Jacobi polynomials to approximate solution of a singular integral equation with Cauchy kernel

Applied Mathematics and Computation ◽

10.1016/j.amc.2006.01.054 ◽

2006 ◽

Vol 181 (1) ◽

pp. 694-707 ◽

Cited By ~ 10

Author(s):

Paweł Karczmarek ◽

Dorota Pylak ◽

Michail A. Sheshko

Keyword(s):

Integral Equation ◽

Approximate Solution ◽

Singular Integral Equation ◽

Singular Integral ◽

Jacobi Polynomials ◽

Cauchy Kernel

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

Georgian Mathematical Journal ◽

10.1515/gmj-2018-0038 ◽

2020 ◽

Vol 27 (1) ◽

pp. 97-102 ◽

Cited By ~ 1

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.

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