Approximate solution of singular integral equations of the first kind with Cauchy kernel

AbstractIn this paper we present a method for the numerical solution of Cauchy type singular integral equations of the first kind on a finite segment which is unbounded at the end points of the segment. Chebyshev polynomials of the first and second kinds are used to derive an approximate solution. Moreover, an estimation error is computed for the approximate solution.

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On Projective Methods of Approximate Solution of Singular Integral Equations

Georgian Mathematical Journal ◽

10.1515/gmj.1996.457 ◽

1996 ◽

Vol 3 (5) ◽

pp. 457-474

Author(s):

A. Jishkariani ◽

G. Khvedelidze

Keyword(s):

Approximate Solution ◽

Rate Of Convergence ◽

Integral Equations ◽

Singular Integral ◽

Singular Integral Equations ◽

Collocation Methods ◽

Projective Methods ◽

Galerkin And Collocation Methods

Abstract The estimate for the rate of convergence of approximate projective methods with one iteration is established for one class of singular integral equations. The Bubnov–Galerkin and collocation methods are investigated.

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Approximate solution of multidimensional singular integral equations

Singular Integral Operators ◽

10.1007/978-3-642-61631-0_18 ◽

1980 ◽

pp. 473-492

Author(s):

Solomon G. Mikhlin ◽

Siegfried Prössdorf

Keyword(s):

Approximate Solution ◽

Integral Equations ◽

Singular Integral ◽

Singular Integral Equations

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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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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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