Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind

Emil Sobhy Shoukralla

doi:doi:10.11648/j.acm.20211003.14

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Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind

Emil Sobhy Shoukralla

Published in Applied and Computational Mathematics (Volume 10, Issue 3)

Received: 15 June 2021 Accepted: 24 June 2021 Published: 30 June 2021

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Abstract

We establish a new straightforward interpolation method for solving linear Volterra integral ‎‎equations with weakly singular kernels. The proposed method is fundamentally different from all other published methods for solving this type of equations. We have modified some vector-matrix barycentric Lagrange interpolation formulas to be convenient for interpolating the kernel twice concerning the two variables of the kernel and introducing new ideas for selecting interpolation nodes that ensure isolation of the singularity of the kernel. We create two rules for selecting the distribution nodes of ‎‎the two kernel variables that do not allow the ‎‎denominator of the kernel to contain an imaginary value. We interpolate the unknown and data functions ‎‎into the corresponding interpolant polynomial; each of the same degree via three matrices, one of ‎‎which is a monomial. By applying the presented method based on the two created rules, we transformed the ‎kernel into a double ‎interpolant polynomial with a degree equal to that of the unknown ‎function via five matrices, two of ‎which are monomials. We substitute the interpolate unknown ‎function twice; on the left side and on the ‎right side of the integral equation to get an ‎algebraic linear system without applying the ‎collocation method. The solution of this system yields ‎the unknown coefficients matrix that is necessary to find the interpolant solution. We ‎solve three ‎different examples for different values of the upper integration variable. The obtained ‎results as ‎shown in tables and figures prove that the obtained interpolate solutions are extraordinarily faster ‎‎to converge to the exact ones using interpolants of lowest degrees and give better results than those obtained by ‎other ‎methods. This confirms the originality and the potential of the presented method.‎

Published in	Applied and Computational Mathematics (Volume 10, Issue 3)
DOI	10.11648/j.acm.20211003.14
Page(s)	76-85
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Lagrange Interpolation, Singular Integral, Weakly Singular Volterra Kernels, Computational Methods, Vandermonde Matrix, Scattering, Radiation, Image Processing

References

[1]	Kendall E. Atkinson: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press (2010).
[2]	Prem K. Kythe and Pratap Puri, Computational Methods for Linear Integral Equations, Birkhäuser, Boston, 2002.
[3]	Abdul-Majid Wazwaz, A First Course in Integral Equations - Solutions Manual, 2nd ed., World Scientific Publishing Co. Pte. Ltd, (2015).
[4]	H. Brunner, Collocation methods for Volterra integral and related functional Equations, Cambridge University Press, Cambridge, 2004.
[5]	E. S. Shoukralla, “A Numerical Method for Solving Fredholm Integral Equations of the First Kind with Logarithmic Kernels and Singular Unknown Functions” Journal of Applied and Computational Mathematics, Springer Nature, (2020) 6: 172.
[6]	E. S. Shoukralla, “Application of Chebyshev Polynomials of the Second Kind to the Numerical Solution of Weakly Singular Fredholm Integral Equations of the First Kind” IAENG International Journal of Applied Mathematics”, Vol. 51, issue. 1, IJAM_51_1_08, (2021).
[7]	E. S. Shoukralla, M. A. Markos, “A new computational method for solving weakly singular Fredholm integral equations of the first kind, IEEE International Conf. on Computer Engineering and Systems (ICCES 2018), Cairo, Egypt, 202-207. 2018.
[8]	E. S. Shoukralla and M. A. Markos, “The economized monic Chebyshev polynomials for solving weakly singular Fredholm integral equations of the first kind,” Asian-European Journal of Mathematics, Vol. 12, No. 1, pp. 2050030-1-2050030-10, October 2018.
[9]	E. S. Shoukralla, “Approximate solution to weakly singular integral equations, Journal of appl. Math Modelling. 20 (1996) 800-803.
[10]	E. S. Shoukralla, “Numerical solution of Helmholtz equation for an open boundary in space, Journal of appl. Math Modeling. 21 (4): 231-232, 1997.
[11]	The Barycentric Lagrange Interpolation via Maclaurin Polynomials for Solving the Second Kind Volterra Integral Equations, Proceedings of 2020 15th IEEE International Conference on Computer Engineering and Systems (ICCES), Ain Shams University, Cairo, Egypt, 2020.
[12]	Multi-techniques method for Solving Volterra Integral Equations of the Second Kind, Proceedings of 2019 14th IEEE International Conference on Computer Engineering and Systems (ICCES), Ain Shams University, Cairo, Egypt, 2019.
[13]	E. S. Shoukralla, H. Elgohary and B. M. Ahmed, “Barycentric ‎Lagrange interpolation for solving Volterra integral equations ‎of the second kind”, Journal of Physics, England, Conference ‎Series, 1447 (2020), 012002.‎
[14]	Jean-Paul Berrut, Lloyd N. Trefethen, Barycentric Lagrange Interpolation, SIAM REVIEW, Society for Industrial and Applied Mathematics, 46 (3) (2004) 501–517.
[15]	Nicholas J. Higham, The numerical stability of barycentric Lagrange interpolation, IMA Journal of Numerical Analysis, 24 (2004) 547-556.
[16]	Zhang Xiao-yong, “Jacobi spectral method for the second-kind Volterra integral equations with a weakly singular kernel”, Applied Mathematical Modelling, Volume 39, Issue 15, 1 August 2015, Pages 4421-4431.
[17]	Can Huang and Martin Stynes, “Spectral Galerkin methods for a weakly singular Volterra integral equation of the second kind”, IMA Journal of Numerical Analysis (2017) 37, 1411–1436.
[18]	Lina Wang, Hongjiong Tian, Lijun Yi, “An hp-version of the discontinuous Galerkin time-stepping method for Volterra integral equations with weakly singular kernels” Applied Numerical Mathematics 161 (2021) 218–232.
[19]	Jingjun Zhao, Teng Long and Yang Xu, “Super Implicit Multistep Collocation Methods for Weakly Singular Volterra Integral Equations”, Numer. Math. Theor. Meth. Appl., Vol. 12, No. 4, pp. 1039-1065, 2019.
[20]	Hossein BEYRAMI and Taher LOTFI,” On the local super convergence of the fully discretized multiprojection method for weakly singular Volterra integral equations of the second kind”, Turkish Journal of Mathematics, (2018) 42: 1400-1423.

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Emil Sobhy Shoukralla. (2021). Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind. Applied and Computational Mathematics, 10(3), 76-85. https://doi.org/10.11648/j.acm.20211003.14

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ACS Style

Emil Sobhy Shoukralla. Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind. Appl. Comput. Math. 2021, 10(3), 76-85. doi: 10.11648/j.acm.20211003.14

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AMA Style

Emil Sobhy Shoukralla. Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind. Appl Comput Math. 2021;10(3):76-85. doi: 10.11648/j.acm.20211003.14

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@article{10.11648/j.acm.20211003.14,
  author = {Emil Sobhy Shoukralla},
  title = {Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind},
  journal = {Applied and Computational Mathematics},
  volume = {10},
  number = {3},
  pages = {76-85},
  doi = {10.11648/j.acm.20211003.14},
  url = {https://doi.org/10.11648/j.acm.20211003.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211003.14},
  abstract = {We establish a new straightforward interpolation method for solving linear Volterra integral ‎‎equations with weakly singular kernels. The proposed method is fundamentally different from all other published methods for solving this type of equations. We have modified some vector-matrix barycentric Lagrange interpolation formulas to be convenient for interpolating the kernel twice concerning the two variables of the kernel and introducing new ideas for selecting interpolation nodes that ensure isolation of the singularity of the kernel. We create two rules for selecting the distribution nodes of ‎‎the two kernel variables that do not allow the ‎‎denominator of the kernel to contain an imaginary value. We interpolate the unknown and data functions ‎‎into the corresponding interpolant polynomial; each of the same degree via three matrices, one of ‎‎which is a monomial. By applying the presented method based on the two created rules, we transformed the ‎kernel into a double ‎interpolant polynomial with a degree equal to that of the unknown ‎function via five matrices, two of ‎which are monomials. We substitute the interpolate unknown ‎function twice; on the left side and on the ‎right side of the integral equation to get an ‎algebraic linear system without applying the ‎collocation method. The solution of this system yields ‎the unknown coefficients matrix that is necessary to find the interpolant solution. We ‎solve three ‎different examples for different values of the upper integration variable. The obtained ‎results as ‎shown in tables and figures prove that the obtained interpolate solutions are extraordinarily faster ‎‎to converge to the exact ones using interpolants of lowest degrees and give better results than those obtained by ‎other ‎methods. This confirms the originality and the potential of the presented method.‎},
 year = {2021}
}

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TY  - JOUR
T1  - Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind
AU  - Emil Sobhy Shoukralla
Y1  - 2021/06/30
PY  - 2021
N1  - https://doi.org/10.11648/j.acm.20211003.14
DO  - 10.11648/j.acm.20211003.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 76
EP  - 85
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20211003.14
AB  - We establish a new straightforward interpolation method for solving linear Volterra integral ‎‎equations with weakly singular kernels. The proposed method is fundamentally different from all other published methods for solving this type of equations. We have modified some vector-matrix barycentric Lagrange interpolation formulas to be convenient for interpolating the kernel twice concerning the two variables of the kernel and introducing new ideas for selecting interpolation nodes that ensure isolation of the singularity of the kernel. We create two rules for selecting the distribution nodes of ‎‎the two kernel variables that do not allow the ‎‎denominator of the kernel to contain an imaginary value. We interpolate the unknown and data functions ‎‎into the corresponding interpolant polynomial; each of the same degree via three matrices, one of ‎‎which is a monomial. By applying the presented method based on the two created rules, we transformed the ‎kernel into a double ‎interpolant polynomial with a degree equal to that of the unknown ‎function via five matrices, two of ‎which are monomials. We substitute the interpolate unknown ‎function twice; on the left side and on the ‎right side of the integral equation to get an ‎algebraic linear system without applying the ‎collocation method. The solution of this system yields ‎the unknown coefficients matrix that is necessary to find the interpolant solution. We ‎solve three ‎different examples for different values of the upper integration variable. The obtained ‎results as ‎shown in tables and figures prove that the obtained interpolate solutions are extraordinarily faster ‎‎to converge to the exact ones using interpolants of lowest degrees and give better results than those obtained by ‎other ‎methods. This confirms the originality and the potential of the presented method.‎
VL  - 10
IS  - 3
ER  -

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Author Information

Emil Sobhy Shoukralla

Department of Engineering Mathematics and Physics, Faculty of Electronic Engineering, Menoufia University, Shibin El Kom, Egypt

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Emil Sobhy Shoukralla. (2021). Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind. Applied and Computational Mathematics, 10(3), 76-85. https://doi.org/10.11648/j.acm.20211003.14

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ACS Style

Emil Sobhy Shoukralla. Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind. Appl. Comput. Math. 2021, 10(3), 76-85. doi: 10.11648/j.acm.20211003.14

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AMA Style

Emil Sobhy Shoukralla. Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind. Appl Comput Math. 2021;10(3):76-85. doi: 10.11648/j.acm.20211003.14

Copy | Download

@article{10.11648/j.acm.20211003.14,
  author = {Emil Sobhy Shoukralla},
  title = {Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind},
  journal = {Applied and Computational Mathematics},
  volume = {10},
  number = {3},
  pages = {76-85},
  doi = {10.11648/j.acm.20211003.14},
  url = {https://doi.org/10.11648/j.acm.20211003.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211003.14},
  abstract = {We establish a new straightforward interpolation method for solving linear Volterra integral ‎‎equations with weakly singular kernels. The proposed method is fundamentally different from all other published methods for solving this type of equations. We have modified some vector-matrix barycentric Lagrange interpolation formulas to be convenient for interpolating the kernel twice concerning the two variables of the kernel and introducing new ideas for selecting interpolation nodes that ensure isolation of the singularity of the kernel. We create two rules for selecting the distribution nodes of ‎‎the two kernel variables that do not allow the ‎‎denominator of the kernel to contain an imaginary value. We interpolate the unknown and data functions ‎‎into the corresponding interpolant polynomial; each of the same degree via three matrices, one of ‎‎which is a monomial. By applying the presented method based on the two created rules, we transformed the ‎kernel into a double ‎interpolant polynomial with a degree equal to that of the unknown ‎function via five matrices, two of ‎which are monomials. We substitute the interpolate unknown ‎function twice; on the left side and on the ‎right side of the integral equation to get an ‎algebraic linear system without applying the ‎collocation method. The solution of this system yields ‎the unknown coefficients matrix that is necessary to find the interpolant solution. We ‎solve three ‎different examples for different values of the upper integration variable. The obtained ‎results as ‎shown in tables and figures prove that the obtained interpolate solutions are extraordinarily faster ‎‎to converge to the exact ones using interpolants of lowest degrees and give better results than those obtained by ‎other ‎methods. This confirms the originality and the potential of the presented method.‎},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind
AU  - Emil Sobhy Shoukralla
Y1  - 2021/06/30
PY  - 2021
N1  - https://doi.org/10.11648/j.acm.20211003.14
DO  - 10.11648/j.acm.20211003.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 76
EP  - 85
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20211003.14
AB  - We establish a new straightforward interpolation method for solving linear Volterra integral ‎‎equations with weakly singular kernels. The proposed method is fundamentally different from all other published methods for solving this type of equations. We have modified some vector-matrix barycentric Lagrange interpolation formulas to be convenient for interpolating the kernel twice concerning the two variables of the kernel and introducing new ideas for selecting interpolation nodes that ensure isolation of the singularity of the kernel. We create two rules for selecting the distribution nodes of ‎‎the two kernel variables that do not allow the ‎‎denominator of the kernel to contain an imaginary value. We interpolate the unknown and data functions ‎‎into the corresponding interpolant polynomial; each of the same degree via three matrices, one of ‎‎which is a monomial. By applying the presented method based on the two created rules, we transformed the ‎kernel into a double ‎interpolant polynomial with a degree equal to that of the unknown ‎function via five matrices, two of ‎which are monomials. We substitute the interpolate unknown ‎function twice; on the left side and on the ‎right side of the integral equation to get an ‎algebraic linear system without applying the ‎collocation method. The solution of this system yields ‎the unknown coefficients matrix that is necessary to find the interpolant solution. We ‎solve three ‎different examples for different values of the upper integration variable. The obtained ‎results as ‎shown in tables and figures prove that the obtained interpolate solutions are extraordinarily faster ‎‎to converge to the exact ones using interpolants of lowest degrees and give better results than those obtained by ‎other ‎methods. This confirms the originality and the potential of the presented method.‎
VL  - 10
IS  - 3
ER  -

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