On the numerical treatment of the singular integral equation of the second kind

Abdou, M. A.; Nasr, Abdurrahman A.

doi:10.1016/s0096-3003(02)00587-8

Cited by 23 publications

(10 citation statements)

References 7 publications

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“…Lifanov in [8] introduced hypersingular integral equations with applications and a numerical solution for a class of these equations of Prandtl's type is given in [6]. Numerical solutions for the Cauchy and Abel type of weakly singular integral equations are discussed in [1][2][3][6][7][8][9][10][11][12][13][14][15][16]. The polar kernel of integral equations has been introduced in [2,16].…”

Section: Introductionmentioning

confidence: 99%

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The approximate solution of a class of Fredholm integral equations with a weakly singular kernel

Babolian

Hajikandi

2011

Journal of Computational and Applied Mathematics

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MSC:Keywords: Cauchy kernel Weakly singular Taylor series Galerkin method Legendre functions a b s t r a c t A method for finding the numerical solution of a weakly singular Fredholm integral equation of the second kind is presented. The Taylor series is used to remove singularity and Legendre polynomials are used as a basis. Furthermore, the Legendre function of the second kind is used to remove singularity in the Cauchy type integral equation. The integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the Cauchy principal value sense without using numerical quadratures. Four examples are given to show the accuracy of the method.

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Section: Introductionmentioning

confidence: 99%

“…One of the weakly singular integral and integro-differential equations with this kernel has been introduced in [9,3,11]. We introduce the following singular integral equation…”

Section: Introductionmentioning

confidence: 99%

The approximate solution of a class of Fredholm integral equations with a weakly singular kernel

Babolian

Hajikandi

2011

Journal of Computational and Applied Mathematics

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“…The integral equations with Cauchy kernel have been widely used in solving problems associated with aerodynamic, hydrodynamic and elasticity (Lifanov, 1996;Ladopoulos, 2000;Abdou and Naser, 2003;Mohankumar and Natarajan, 2008;Lara and Mariagrazia, 2005;Kasozi and Paulsen, 2005a;Kasozi and Paulsen, 2005b;Ganji et al, 2008;Thukral, 2005).…”

Section: Introductionmentioning

confidence: 99%

A Note on the Numerical Solution for Fredholm Integral Equation of the Second Kind with Cauchy kernel

Abdulkawi¹,

Long²,

Eshkuvatov³

2011

Journal of Mathematics and Statistics

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“…They showed that the approximate method is exact when the force function f (t) is linear. Abdou and Naser [1] considered CSIEs of the second kind and used orthogonal polynomials (Legendre) to find the approximate solution, and described the physical meaning of the stated problems. Rashed [10] presented two numerical methods for solution of non-singular integral equations of the first kind using Chebyshev polynomials of the first kind to approximate the kernel and unknown function.…”

Section: Introductionmentioning

confidence: 99%

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

Eshkuvatov

Long

Abdulkawi

2009

Applied Mathematics Letters

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a b s t r a c tIn this work a study of efficient approximate methods for solving the Cauchy type singular integral equations (CSIEs) of the first kind, over a finite interval, is presented. In the solution, Chebyshev polynomials of the first kind, T n (x), second kind, U n (x), third kind, V n (x), and fourth kind, W n (x), corresponding to respective weight functions1 2 , have been used to obtain systems of linear algebraic equations. These systems are solved numerically. It is shown that for a linear force function the method of approximate solution gives an exact solution, and it cannot be generalized to any polynomial of degree n. Numerical results for other force functions are given to illustrate the efficiency and accuracy of the method.

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On the numerical treatment of the singular integral equation of the second kind

Cited by 23 publications

References 7 publications

The approximate solution of a class of Fredholm integral equations with a weakly singular kernel

The approximate solution of a class of Fredholm integral equations with a weakly singular kernel

A Note on the Numerical Solution for Fredholm Integral Equation of the Second Kind with Cauchy kernel

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

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