A Computational Method for Solving Fredholm Integral Equations of the Second Kind

Document Type : Original Article

Authors

1 Deptartment.of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Minufiya University Menouf, Egypt

2 Deptartment..of Physics and Engineering Mathematics, Faculty of Engineering and Technology., GUC in Egypt Cairo, Egypt

Abstract

this paper presents a new double­-approximate computational method for solving Fredholm integral equations of the second kind. The method is based on the approximation of the given data function, and the unknown function by utilizing Legendre polynomials in matrix form, while the approximate function of Legendre coefficients of the first approximation of the kernel is expanded once again using Legendre polynomials. In addition, the double substitution of the unknown function into the considered integral equation provides access to a linear system of an algebraic equation, which is equivalent to the required solution. Convergence in the mean of the unknown function is proved. Finally, five illustrative examples are presented. It turns out that the obtained approximate solutions converge to the exact solutions that demonstrate the authenticity and the efficiency of the new method. 

Keywords

References

[1]     Olha Ivanyshyn Yaman, Gazi Aldemir, “Boundary integral equations for the exterior Robin problem in two dimensions”, Applied Mathematics and Computation, Vol. 337, pp. 25-33, 2018.
[2]     Xavier Claeysa, Ralf Hiptmairb, Elke, Spindlerb, “Second-kind boundary integral equations for electromagnetic scattering at composite objects” Computers & Mathematics with Applications, Vol. 74, Issue 11, pp. 2650-2670, 2017.
[3]     P. Kythe and P. Puri, Computational methods for linear integral equations: Springer Science & Business Media, 2011.
[4]     K. E. Atkinson, "The numerical solution of integral equation of the second kind," ed: Cambridge University Press, Cambridge, 1996.
[5]     A. M. Wazwaz, A first course in integral equations: World Scientific Publishing Company, 2015.
[6]     M. Rahman, Integral equations and their applications: WIT press, 2007.
[7]     Kress, Rainer. Linear Integral Equations. New York: Springer-Verlag, 1989.
[8]     R. P. Kanwal, Linear integral equations: Springer Science & Business Media, 2013.
[9]     M. Krasnov, A. Kiselev, G. Makarenko. Problems and Exercises In Integral Equations. Moscow: Mir Publishers, 1971.
[10]  S. S. Ray and P. K. Sahu, "Numerical Methods for Solving Fredholm Integral Equations of Second Kind," Abstract and Applied Analysis, vol. 2013, pp. 1-17, 2013.
[11]  B. Yilmaz and Y. Cetin, "Numerical solutions of the Fredholm integral equations of the second type," BISKA, New Trends in Mathematical Science, vol. 3, pp. 284-292, 2017.
[12]  S. Panda, S. C. Martha, and A. Chakrabarti, "A modified approach to numerical solution of Fredholm integral equations of the second kind," Applied Mathematics and Computation, vol. 271, pp. 102-112, 2015.
[13]  J. Rashidinia and M. Zarebnia, "Numerical solution of linear integral equations by using Sinc–collocation method," Applied Mathematics and Computation, vol. 168, pp. 806-822, 2005.
[14]  X.-Z. Liang, M.-C. Liu and X.-J. Che, "Solving second kind integral equations by Galerkin methods with continuous orthogonal wavelets," Journal of computational and applied mathematics, vol. 136, pp. 149-161, 2001
[15]  A. Chakrabarti and S. C. Martha, "Approximate solutions of Fredholm integral equations of the second kind," Applied Mathematics and Computation, vol. 211, pp. 459-466, 2009.
[16]  X.-C. Zhong, "A new Nyström-type method for Fredholm integral equations of the second kind," Applied Mathematics and Computation, vol. 219, pp. 8842-8847, 2013.
[17]  E. S. Shoukralla," Approximate Solution of Fredholm Integral Equations of the second kind using a combine method, ’Mathematics and Computer Science, vol. 42, pp. 42-52, 2016.
[18]  E. S. Shoukralla," A matrix iterative technique for the solution of Fredholm Integral equations of the second kind, ’Electronic Journal of Mathematical Analysis and Applications, (EJMAA, vol. 4, 2016.
[19]  Mauro M. Doria “Chebyshev, Legendre, Hermit and other orthogonal polynomials in D dimensions”, reports on mathematical physics, Vol. 81, 2018.
[20]  R. Piessens, "Computing integral transforms and solving integral equations using Chebyshev polynomial approximations," Journal of Computational and Applied Mathematics, vol. 121, pp. 113-124, 2000.
[21]  N. Patanarapeelert and V. Varnasavang, "Comparison Study of Series Approximation and Convergence between Chebyshev and Legendre Series," Applied Mathematical Sciences, vol. 7, pp. 3225-3237, 2013.
[22]  E. S. Shoukralla, "Double – Shifted Legendre Polynomials for the solution of Fredholm Integral Equation of the Second Kind” Electronic Engineering Bulletin, Egypt, No. 13, pp. 1-5, 1997.
[23]  S. Yalçınbas, M. Aynigül, and T. Akkaya, "Legendre series solutions of Fredholm integral equations," Mathematical and Computational Applications, vol. 15, pp. 371-381, 2010.
[24]  W. W. Bell, Special functions for scientists and engineers: Courier Corporation, 2004.
[25]  H. Wang and S. Xiang, "On the convergence rates of Legendre approximation," Mathematics of Computation, vol. 81, pp. 861-877, 2012.
[26]  K. Maleknejad, K. Nouri, and M. Yousefi, "Discussion on convergence of Legendre polynomial for numerical solution of integral equations," Applied Mathematics and Computation, vol. 193, pp. 335-339, 2007.
[27]  Abigail I. Fairbairn, Mark A. Kelmanson, “Spectrally accurate Nyström-solver error bounds for 1-D Fredholm integral equations of the second kind” Applied Mathematics and Computation, Vol. 315, pp. 211-223, 2017.
[28]  R. Debbar, H. Guebbai, and Z. Zereg, "Improving the convergence order of the regularization method for Fredholm integral equations of the second kind," Applied Mathematics and Computation, vol. 289, pp. 204-213, 2016.
[29]  Z. Cvetkovski, Inequalities: theorems, Techniques, and selected problems: Springer Science & Business Media, 2012.
 
Menoufia Journal of Electronic Engineering Research
Volume 28, ICEEM2019-Special Issue
ICEEM2019-Special Issue: 1st International Conference on Electronic Eng., Faculty of Electronic Eng., Menouf, Egypt, 7-8 Dec.
2019
Pages 280-285

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