Approximate Solution of a Singular Integral Cauchy-Kernel Equation of the First Kind

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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Orthogonal Approximate Solution of Cauchy-type Singular Integral Equations

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2003-0021 ◽

2003 ◽

Vol 3 (2) ◽

pp. 330-356

Author(s):

R. Smarzewski ◽

M. A. Sheshko

Keyword(s):

Approximate Solution ◽

Integral Equations ◽

Singular Integral ◽

Chebyshev Polynomials ◽

Singular Integral Equations ◽

Approximate Solutions ◽

Cauchy Type

AbstractChebyshev polynomials of the first and second kind are used to derive approximate solutions of the Cauchy-type singular integral equations.

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A method of numerical solution of cauchy-type singular integral equations with generalized kernels and arbitrary complex singularities

Journal of Computational Physics ◽

10.1016/0021-9991(79)90117-7 ◽

1979 ◽

Vol 30 (3) ◽

pp. 309-323 ◽

Cited By ~ 26

Author(s):

Pericles S Theocaris ◽

Nikolaos I Ioakimidis

Keyword(s):

Numerical Solution ◽

Integral Equations ◽

Singular Integral ◽

Singular Integral Equations ◽

Cauchy Type ◽

Complex Singularities ◽

Arbitrary Complex

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

Applied Mathematics Letters ◽

10.1016/j.aml.2008.08.001 ◽

2009 ◽

Vol 22 (5) ◽

pp. 651-657 ◽

Cited By ~ 28

Author(s):

Z.K. Eshkuvatov ◽

N.M.A. Nik Long ◽

M. Abdulkawi

Keyword(s):

Approximate Solution ◽

Integral Equations ◽

Singular Integral ◽

Singular Integral Equations ◽

Cauchy Kernel

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Application of Faber polynomials to the approximate solution of singular integral equations with the Cauchy kernel

Differential Equations ◽

10.1134/s0012266113020067 ◽

2013 ◽

Vol 49 (2) ◽

pp. 198-209 ◽

Cited By ~ 13

Author(s):

P. Wójcik ◽

M. A. Sheshko ◽

S. M. Sheshko

Keyword(s):

Approximate Solution ◽

Integral Equations ◽

Singular Integral ◽

Singular Integral Equations ◽

Faber Polynomials ◽

Cauchy Kernel

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Application of the Gauss-Laguerre and Radau-Laguerre quadrature rules to the numerical solution of cauchy type singular integral equations

Computers & Structures ◽

10.1016/0045-7949(81)90084-5 ◽

1981 ◽

Vol 14 (1-2) ◽

pp. 63-70 ◽

Cited By ~ 3

Author(s):

Nikolaos I. Ioakimidis

Keyword(s):

Numerical Solution ◽

Integral Equations ◽

Singular Integral ◽

Singular Integral Equations ◽

Quadrature Rules ◽

Cauchy Type

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On the numerical solution of Cauchy type singular integral equations by the collocation method

Applied Mathematics and Computation ◽

10.1016/0096-3003(83)90041-3 ◽

1983 ◽

Vol 12 (1) ◽

pp. 49-60 ◽

Cited By ~ 2

Author(s):

N.I. Ioakimidis

Keyword(s):

Numerical Solution ◽

Integral Equations ◽

Collocation Method ◽

Singular Integral ◽

Singular Integral Equations ◽

Cauchy Type

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Uniform Convergence of a Collocation Method for the Numerical Solution of Cauchy-Type Singular Integral Equations: A Generalization

IMA Journal of Numerical Analysis ◽

10.1093/imanum/12.1.31 ◽

1992 ◽

Vol 12 (1) ◽

pp. 31-45 ◽

Cited By ~ 7

Author(s):

J. A. CUMINATO

Keyword(s):

Uniform Convergence ◽

Numerical Solution ◽

Integral Equations ◽

Collocation Method ◽

Singular Integral ◽

Singular Integral Equations ◽

Cauchy Type

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On the numerical solution of Cauchy type singular integral equations and the determination of stress intensity factors in case of complex singularities

Zeitschrift für angewandte Mathematik und Physik ◽

10.1007/bf01601675 ◽

1977 ◽

Vol 28 (6) ◽

pp. 1085-1098 ◽

Cited By ~ 23

Author(s):

P. S. Theocaris ◽

N. I. Ioakimidis

Keyword(s):

Stress Intensity ◽

Numerical Solution ◽

Integral Equations ◽

Singular Integral ◽

Stress Intensity Factors ◽

Singular Integral Equations ◽

Cauchy Type ◽

Intensity Factors ◽

Complex Singularities

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Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function

Mathematics ◽

10.3390/math7100872 ◽

2019 ◽

Vol 7 (10) ◽

pp. 872 ◽

Cited By ~ 3

Author(s):

◽

Shuhuang Xiang ◽

Guidong Liu

Keyword(s):

Integral Equation ◽

Singular Integral Equation ◽

Numerical Solution ◽

Integral Equations ◽

Kernel Function ◽

Singular Integral ◽

Singular Integral Equations ◽

Cauchy Type ◽

Error Analyses ◽

Highly Oscillatory

This paper aims to present a Clenshaw–Curtis–Filon quadrature to approximate thesolution of various cases of Cauchy-type singular integral equations (CSIEs) of the second kind witha highly oscillatory kernel function. We adduce that the zero case oscillation (k = 0) proposed methodgives more accurate results than the scheme introduced in Dezhbord at el. (2016) and Eshkuvatovat el. (2009) for small values of N. Finally, this paper illustrates some error analyses and numericalresults for CSIEs.

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