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

Karczmarek, Paweł; Pylak, Dorota; Sheshko, M. A.

doi:10.1016/j.amc.2006.01.054

Cited by 17 publications

(13 citation statements)

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“…So many different methods have been developed to obtain an approximate solution of a Cauchy integral equation such as iteration method [7], Berstein polynomials method [8], Jacobi polynomials method [9], Cubic spline method [10], rational functions method [11], generalized inverses method [12] and a polynomial expansion for the unknown by Mohankumar and Natarajan [13]. Bonis and Laurita [14] have proposed a Nyström method to approximate the solutions of Cauchy singular integral equations with constant coefficients having a negative index.…”

Section: Introductionmentioning

confidence: 99%

A new efficient method for cases of the singular integral equation of the first kind

Lotfi

Mahdiani

2016

Journal of Computational and Applied Mathematics

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Various cases of Cauchy type singular integral equation of the first kind occur rather frequently in mathematical physics and possess very unusual properties. These equations are usually difficult to solve analytically, and it is required to obtain approximate solutions. This paper investigates the numerical solution of various cases of Cauchy type singular integral equations using reproducing kernel Hilbert space (RKHS) method. The solution u(x) is represented in the form of a series in the reproducing kernel space, afterwards the n-term approximate solution u n (x) is obtained and it is proved to converge to the exact solution u(x). The major advantage of the method is that it can produce good globally smooth approximate solutions. Moreover, in this paper, an efficient error estimation of the RKHS method is introduced. Finally, numerical experiments show that our reproducing kernel method is efficient.

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Section: Introductionmentioning

confidence: 99%

A new efficient method for cases of the singular integral equation of the first kind

Lotfi

Mahdiani

2016

Journal of Computational and Applied Mathematics

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“…Boundary element method and finite element method are intensively eminent numerical approaches to evaluate partial differential equations (PDEs), which appear in variety of disciplines from engineering to astronomy and quantum mechanics [1][2][3][4][5]. Although these methods lead PDEs to Fredholm integral equations or Voltera integral equations, but these kind of integral equations posses integrals of oscillatory, Cauchy-singular, logarithmic singular, weak singular kernel functions.…”

Section: Introductionmentioning

confidence: 99%

“…where t ∈ (−1, 1), k ≫ 1, α ∈ [−1, 1], f (x) is relatively smooth function. For integral (1) the developed strategy for logarithmic singularity log(x − α) is valid for α ∈ [−1, 1]. In particular, the highly oscillatory integral, ∫ 1 −1 f (x)e ikx dx has been computed by many methods such as asymptotic expansion, Filon method,…”

Section: Introductionmentioning

confidence: 99%

Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels

Saira

Xiang

2019

Symmetry

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In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, ⨍ − 1 1 f ( x ) log ( x − α ) e i k x x − t d x , t ∉ ( − 1 , 1 ) , α ∈ [ − 1 , 1 ] for a smooth function f ( x ) . This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples.

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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%

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

Cited by 17 publications

References 1 publication

A new efficient method for cases of the singular integral equation of the first kind

A new efficient method for cases of the singular integral equation of the first kind

Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels

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

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