An approximate solution of a complete linear singular integral equation with the Hilbert 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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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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Numerical solution of a weakly singular integral equation by the method of orthogonal polynomials

Journal of the Belarusian State University. Mathematics and Informatics ◽

10.33581/2520-6508-2021-3-98-103 ◽

2021 ◽

pp. 98-103

Author(s):

Sergei M. Sheshko

Keyword(s):

Integral Equation ◽

Approximate Solution ◽

Singular Integral Equation ◽

Orthogonal Polynomials ◽

Numerical Solution ◽

Singular Integral ◽

Algebraic Equations ◽

Linear Algebraic Equations ◽

Expansion Coefficients ◽

Analytical Expressions

A scheme is constructed for the numerical solution of a singular integral equation with a logarithmic kernel by the method of orthogonal polynomials. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

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