Shifted Legendre Polynomials For Solving Second Kind Fredholm Integral Equations | ||||
| Menoufia Journal of Electronic Engineering Research | ||||
| Article 10, Volume 30, Issue 1, January 2021, Page 76-83 PDF (532.47 K) | ||||
| Document Type: Original Article | ||||
| DOI: 10.21608/mjeer.2021.146279 | ||||
 View on SCiNiTO | ||||
| Authors | ||||
| Shoukralla, E.. S1; Elgohary, . H1; Morgan. , M2 | ||||
| 1Department of Physics and EngineeringMathematics, Faculty of Electronic Engineering, Menoufia University Menouf, Egypt | ||||
| 2Department of Physics and Engineering Mathematics, Faculty of Engineering and Technology, German University, Egypt Cairo, Egypt | ||||
| Abstract | ||||
| —in this paper, present a computational method for solving Fredholm integral equations of the second kind. The method based on the application of the shifted Legendre polynomials in matrix forms. We create a technique for extracting the Legendre coefficients of each polynomial away so that each Legendre polynomial is rewritten in the form of its coefficient’s matrix multiplied by the monomial basis function matrix. This technique significantly reduces the round off errors. By using this technique, the unknown and the data functions are expressed in two forms; each consists of three matrices. The kernel is approximated twice relevant to its two variables so that it is transformed into a form consists of a product of five matrices. By substituting by the approximate unknown function into the left and the right sides of the integral equation, we obtain an algebraic linear system of the equations without applying the collocation points. Moreover, we adapted the Gauss–Quadrature rule in an adjustment form and applied it for computing the resulted integrals. The convergence in the mean of the approximate solution and the kernel are proved. Additionally, the maximum norm error is studied, and it is found equal to zero. Numerical results are obtained for five examples to clarify the simplicity, efficiency, and reliability of the method. The obtained solutions are equal or rapidly converge to the exact solutions. | ||||
| Keywords | ||||
| —Fredholm integral equations; Legendre polynomials; Computational methods; approximate solution; error estimation | ||||
| References | ||||
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[1] P. Kythe and P. Puri, Computational methods for linear integral equations: Springer Science & Business Media, 2011. [2] K. E. Atkinson, "The numerical solution of integral equation of the second kind," ed: Cambridge University Press, Cambridge, 1996. [3] R. P. Kanwal, Linear integral equations: Springer Science & Business Media, 2013. [4] Xavier Claeys, Ralf Hiptmair , Elke Spindler. "Second-kind boundary integral equations for electromagnetic scattering at composite objects." Computers and Mathematics with Applications 74 (December 2017): 2650–2670. [5] Gomez, Luis J., and Abdulkadir C. Yücel. "The ICVSIE: A General Purpose Integral Equation Method for Bio- Electromagnetic Analysis." IEEE Transactions on Biomedical Engineering . IEEE, 2018. 565 - 574. [6] Kechen Wang, Mengmeng Li, Dazhi Ding, Rushan Chen. "A parallelizable direct solution of integral equation methods for electromagnetic analysis." Engineering Analysis with Boundary Elements 85 (December 2017): 158–164. [7] E. S. Shoukralla ―Numerical Solution of Helmholtz Equation for an Open — Boundary in space‖, J. Applied. Mathematical. Modelling, Elsevier, New York, Vol. 21, 231-232, April 1997. [8] E. S. Shoukralla, ―Approximate solution to weakly singular integral equations‖, Journal of appl. Mathematical. Modelling,. 20 (1996) 800-803. [9] 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, 2020. [10] E. S. Shoukralla, M. Kamel, and M. A. Markos, ―A new computational method for solving weakly singular Fredholm integral equations of the first kind‖ Published in the 13th IEEE International Conference on Computer Engineering and Systems (ICCES 2018). Cairo, Egypt on December, (IEEE Xplore) 2018. [11] E. S. Shoukralla and M. A. Markos, ―Numerical Solution of a Certain Class of Singular Fredholm Integral Equations of the First Kind via the Vandermonde Matrix‖, International Journal of Mathematical Models and Methods in Applied Science‖, Vol. 14, 2020, 48-53. [12] E. S. Shoukralla, and B. M. Ahmed. "Multi-techniques method for Solving Volterra Integral Equations of the Second Kind." 2019 14th International Conference on Computer Engineering and Systems (ICCES). IEEE, 2019. [13] E. S. Shoukralla, H. Elgohary and B. M. Ahmed, ―Barycentric Lagrange interpolation for solving Volterra integral equations of the second kind‖ Published in the 4th International Conference on Advanced Technology and Applied Sciences (ICaTAS2019), Journal of Physics, England, Conference Series, 1447(2020),012002. [14] E. S. Shoukralla and B. M. Ahmed, ―Numerical Solutions of Volterra Integral Equations of the Second Kind using Lagrange interpolation via the Vandermonde matrix‖ 4th International Conference on Advanced Technology and Applied Sciences (ICaTAS’2019), Journal of Physics, England, Conference Series, 1447(2020), 012003. 8 3 [15] Y.Liu. "Application of the Chebyshev polynomial in solving Fredholm integral equations." Mathematical and Computer Modelling 50, no. 3-4 (August 2009): 465-469. [16] M. M. Doria, R. C. V. COELHO ―Chebyshev, Legendre, Hermit and other orthogonal polynomials in D dimensions‖, Reports on Mathematical Physics, Vol. 81, 2018. [17] Zaheer-ud-Din, Siraj-ul-Islam. "Meshless methods for one-dimensional oscillatory Fredholm integral equations." Applied Mathematics and Computation 324 (May 2018): 156-173. [18] D. Barrera, F. Elmokhtari, D. Sbibih. "Two methods based on bivariate spline quasi-interpolants for solving Fredholm integral." Applied Numerical Mathematics 127 (May 2018): 78-94. [19] Guangqing Long ; Lifeng Xuan ; Jianjun Chen. "Extrapolation of Discrete Multi-projection Methods for Fredholm Integral Equations of the Second Kind." Computational Intelligence and Security. Hong Kong, China: IEEE, 2018. [20] Ahmet Alttrk, Serpil ahin "An Application of the Weighted Mean Value Method to Fredholm integral equations withToeplitz plus Hankel kernels." Journal of Interpolation and Approximation in Scientific Computing, no. 2 (June 2017): 9-17. [21] Jalil Talab Abdullah, and Ali Hussein Shuaa Al-Taie, ―A comparison of Numerical Solutions for Linear Fredholm Integral Equation of the Second Kind‖ First International Scientific Conference Al-Ayen University, IOP Conf. Series: Journal of Physics: Conf. Series 1279 (2019)012067. [22] Srikumar Panda, S.C.Martha,A.Chakrabarti ―Amodified approachto numerical solution of Fredholm integral equations of the second kind‖, Applied Mathematics and Computation 271 (2015) 102–112 [23] Mutaz Mohammad,‖ A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle, symmetry MDPI , 2019, 11, 854. [24] K.Maleknejad, Y. Mahmoudi ―Numerical solution of linear Fredholm integral equation by using hybrid Taylor and Block-Pulse functions‖,Applied Mathematics and Computation. 149 (2004) 799–806. [25] K. Maleknejad, K. Nouri, M. Yousefi. "Discussion on convergence of Legendre polynomial for numerical solution of integral equations." Applied Mathematics and Computation (Elsevier) 193, no. 2 (November 2007): 335-339. [26] C. Allouch, A. Boujraf, M. Tahrichi. "Discrete superconvergent degenerate kernel method forFredholmintegral equations." Mathematics and Computers in Simulation (Elsevier) 164 (October 2019): 24-32. [27] Z. Cvetkovski, Inequalities: Theorems, Techniques, and Selected Problems: Springer Science & Business Media, 2012. [28] Fabio Silva Botelho, ―Real Analysis and Applications‖, Springer International Publishing AG, part of Springer Nature 2018. | ||||
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