Numerical solution of certain Cauchy singular integral equations using a collocation scheme

The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the numerical approximation solution of CSIEs, the strong singularity and accuracy of the numeri...

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Published inAdvances in difference equations Vol. 2020; no. 1; pp. 1 - 15
Main Author Seifi, Ali
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LanguageEnglish
Published Cham Springer International Publishing 29.09.2020
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Abstract The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the numerical approximation solution of CSIEs, the strong singularity and accuracy of the numerical methods are still two important challenges for these integral equations. In this paper, we focus on the smooth transformation and implementation of Bessel basis polynomials (BBP). The reduction of the CSIEs-2 into a system of algebraic equations with the Gauss–Legendre collocation points simplifies this technique. The technique of performing numerical approximation of the solution is well presented and illustrated in the matrix form. Also, the convergence and error bound associated with the scheme are established. Finally, several experiments show the reliability and numerical efficiency of the proposed scheme in comparison with other methods.
AbstractList The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the numerical approximation solution of CSIEs, the strong singularity and accuracy of the numerical methods are still two important challenges for these integral equations. In this paper, we focus on the smooth transformation and implementation of Bessel basis polynomials (BBP). The reduction of the CSIEs-2 into a system of algebraic equations with the Gauss–Legendre collocation points simplifies this technique. The technique of performing numerical approximation of the solution is well presented and illustrated in the matrix form. Also, the convergence and error bound associated with the scheme are established. Finally, several experiments show the reliability and numerical efficiency of the proposed scheme in comparison with other methods.
Abstract The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the numerical approximation solution of CSIEs, the strong singularity and accuracy of the numerical methods are still two important challenges for these integral equations. In this paper, we focus on the smooth transformation and implementation of Bessel basis polynomials (BBP). The reduction of the CSIEs-2 into a system of algebraic equations with the Gauss–Legendre collocation points simplifies this technique. The technique of performing numerical approximation of the solution is well presented and illustrated in the matrix form. Also, the convergence and error bound associated with the scheme are established. Finally, several experiments show the reliability and numerical efficiency of the proposed scheme in comparison with other methods.
ArticleNumber 537
Author Seifi, Ali
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  organization: Department of Mathematics, Hamedan Branch, Islamic Azad University
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Cites_doi 10.1016/S0096-3003(02)00587-8
10.1016/0022-247X(84)90098-2
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Keywords Bessel polynomial
Collocation scheme
Cauchy singular integral equation
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Snippet The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2)....
Abstract The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind...
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SubjectTerms Analysis
Applications
Bessel polynomial
Cauchy singular integral equation
Collocation scheme
Difference and Functional Equations
Functional Analysis
Mathematics
Mathematics and Statistics
Methods
Ordinary Differential Equations
Partial Differential Equations
Topics in Special Functions and q-Special Functions: Theory
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Title Numerical solution of certain Cauchy singular integral equations using a collocation scheme
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