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 in | Advances in difference equations Vol. 2020; no. 1; pp. 1 - 15 |
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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. |
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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 |
Author_xml | – sequence: 1 givenname: Ali orcidid: 0000-0001-5422-9554 surname: Seifi fullname: Seifi, Ali email: aliseifi.math@yahoo.com, aliseifi.math@gmail.com 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 10.1016/j.cam.2016.01.011 10.1016/j.chaos.2019.109467 10.1016/j.engfracmech.2007.04.024 10.1515/zna-2010-8-908 10.3844/jmssp.2011.68.72 10.1016/0021-9045(82)90040-5 10.1016/j.cam.2018.11.020 10.1016/j.apnum.2011.12.008 10.1016/j.cam.2012.09.028 10.1016/j.amc.2012.07.017 10.1016/j.cam.2009.09.034 10.1016/0377-0427(90)90363-5 10.1016/j.cam.2014.09.030 10.1016/j.amc.2006.01.054 10.1201/9780203490303 10.1016/j.amc.2012.11.034 10.1016/j.chaos.2020.109811 10.1007/978-94-015-8648-1 10.1007/s00500-019-04031-1 10.1016/j.chaos.2020.109619 10.1002/mma.6297 10.1016/S0377-0427(97)00045-9 10.1016/0022-247X(83)90160-9 10.1186/s13662-017-1339-3 10.1002/mma.6335 10.1016/j.cam.2009.07.034 |
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Kumar – volume-title: Solution of Crack Problems year: 1996 ident: 2996_CR7 doi: 10.1007/978-94-015-8648-1 contributor: fullname: D.A. Hills – volume: 352 start-page: 50 year: 2019 ident: 2996_CR24 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2018.11.020 contributor: fullname: L. Yang – volume: 219 start-page: 1108 year: 2012 ident: 2996_CR25 publication-title: Appl. Math. Comput. doi: 10.1016/j.amc.2012.07.017 contributor: fullname: A. Mennouni – volume: 43 start-page: 5564 year: 2020 ident: 2996_CR4 publication-title: Math. Methods Appl. Sci. doi: 10.1002/mma.6297 contributor: fullname: S. Kumar – volume: 146 start-page: 373 year: 2003 ident: 2996_CR6 publication-title: Appl. Math. Comput. doi: 10.1016/S0096-3003(02)00587-8 contributor: fullname: M.A. Abdou – volume: 82 start-page: 93 year: 1997 ident: 2996_CR22 publication-title: J. Comput. Appl. Math. doi: 10.1016/S0377-0427(97)00045-9 contributor: fullname: K. Diethelm – volume-title: Handbook of Computational Methods for Integration year: 2004 ident: 2996_CR27 doi: 10.1201/9780203490303 contributor: fullname: P. Kythe |
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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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