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Investigation of the Numerical Solution of Integral Equation with Kernels Involving Logarithmic Functions

Received: 21 November 2016     Accepted: 30 November 2016     Published: 3 January 2017
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Abstract

The main aim of this paper is to investigate the numerical solution of first kind integral equation of logarithmic kernel when using spectral method. Our approach consists of limiting the boundary to the unit interval and specify a logarithmic kernel. The behavior of the solution on the unit interval was analyzed and the advantages and disadvantages of this approach was shown.

Published in Mathematics and Computer Science (Volume 2, Issue 1)
DOI 10.11648/j.mcs.20170201.11
Page(s) 1-5
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), 2017. Published by Science Publishing Group

Keywords

Integral Equation, Spectral Method, Logarithmic Kernel

References
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[8] Canuto C., Hussaini M. Y., Quarteroni A., and Zang T. A. (2006) Spectral Methods. Fundamentals in Single Domains. Springer-Verlag, Berlin Heidelberg
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[10] E. S. Shoukralla, S. A. EL-Serafi and Nermein. Saber, “A Modified Iterative Method for the Solution of Fredholm Integral Equation of the Second Kind Via Matrices” International Journal of Universal Mathematics and Mathematical Sciences, 01 Issues No.: 02, ISSN No.: 2454-7271, Dec 2015.
[11] G. C. Hsiao, W. L. Wendland, Boundary Integral Equations, Springer, 2008.
[12] G. H. Hardy and J. E. Littlewood, Some new properties of Fourier constants, in Collected Papers of G. H. Hardy III, Oxford University Press (1969), 348-399.
[13] G. Fairweather and R. L. Johnston, The method of fundamental solutions for problems in potential theory in Treatment of Integral Equations by Numerical Methods (Ed. by C. T. H. Baker and G. F. Miller), Academic Press Inc. (London) Ltd.
[14] J. CF. Telles and L. C. Wrobel, Boundary Element Techniques, Springer-Verlag, 1984.
[15] Jafarian A, Esmailzadeh Z, Khoshbakhti L. A numerical method for solving nonlinear integral equations in the Urysohn form. Applied Mathematical Sciences. 2013; 7 (28): 1375–1385.
[16] John P. Boyd, Chebyshev and Fourier Spectral Methods, DOVER Publications (2000), Inc. 31 East 2nd Street Mineola, New York 11501.
[17] M. JLELI, M. MURSALEEN, B. SAMET, Q-Integral Equations of Fractional Orders, Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 17, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu
[18] M. A. Jaswon and G. T. Symm, Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, London 1977.
[19] Maleknejad K, Derili H. The collocation method for Hammerstein equations by Daubechies wavelets. Applied Mathematics and Computation. 2006; 172 (2): 846–864.
[20] Shoukralla, E. S., EL-Serafi, S. A., Saber, Nermein A, "A matrix Iterative technique for the solution of Fredholm Integral Equations of the second kind" "Electronic Journal of Mathematical Analysis and Applications" Vol. 4 (1) Jan. 2016, pp. 192-196. ISSN: 2090-729.
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    Azhari Ahmad Abdalla. (2017). Investigation of the Numerical Solution of Integral Equation with Kernels Involving Logarithmic Functions. Mathematics and Computer Science, 2(1), 1-5. https://doi.org/10.11648/j.mcs.20170201.11

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    ACS Style

    Azhari Ahmad Abdalla. Investigation of the Numerical Solution of Integral Equation with Kernels Involving Logarithmic Functions. Math. Comput. Sci. 2017, 2(1), 1-5. doi: 10.11648/j.mcs.20170201.11

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    AMA Style

    Azhari Ahmad Abdalla. Investigation of the Numerical Solution of Integral Equation with Kernels Involving Logarithmic Functions. Math Comput Sci. 2017;2(1):1-5. doi: 10.11648/j.mcs.20170201.11

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  • @article{10.11648/j.mcs.20170201.11,
      author = {Azhari Ahmad Abdalla},
      title = {Investigation of the Numerical Solution of Integral Equation with Kernels Involving Logarithmic Functions},
      journal = {Mathematics and Computer Science},
      volume = {2},
      number = {1},
      pages = {1-5},
      doi = {10.11648/j.mcs.20170201.11},
      url = {https://doi.org/10.11648/j.mcs.20170201.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20170201.11},
      abstract = {The main aim of this paper is to investigate the numerical solution of first kind integral equation of logarithmic kernel when using spectral method. Our approach consists of limiting the boundary to the unit interval and specify a logarithmic kernel. The behavior of the solution on the unit interval was analyzed and the advantages and disadvantages of this approach was shown.},
     year = {2017}
    }
    

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    T2  - Mathematics and Computer Science
    JF  - Mathematics and Computer Science
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    AB  - The main aim of this paper is to investigate the numerical solution of first kind integral equation of logarithmic kernel when using spectral method. Our approach consists of limiting the boundary to the unit interval and specify a logarithmic kernel. The behavior of the solution on the unit interval was analyzed and the advantages and disadvantages of this approach was shown.
    VL  - 2
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Author Information
  • Department of Mathematics, University of Hail, Hail, Saudi Arabia

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