International Journal of Nonlinear Analysis and Applications
High performance of the Remez algorithm in finding polynomial approximations for the solution of integral equations
Document Type : Research Paper
Author
Department of Mathematics, Higher Education Complex of Bam, Bam, Iran.
Abstract
In this paper, the solution of the general integral equation of the second kind is approximated using polynomials. These polynomials are obtained based on the Remez algorithm and the minimization of residual function. The nature of the use of the Remez algorithm in the proposed method will lead to the conversion of the integral equation to a system of algebraic equations and obtaining the best polynomial approximation for the solution of an integral equation. Also, the convergence analysis of the approach is discussed. Finally, some numerical examples and comparisons with previous results confirm the efficiency and high accuracy of the presented method.
Keywords
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[7] N. Negarchi and K. Nouri, Numerical solution of Volterra– Fredholm integral equations using the collocation method based on a special form of the M¨untz– Legendre polynomials, J. Comp. Appl. Math. 344 (2018), 15–24.
[8] S. Yal¸cinbas, M. Sezer and H.H. Sorkun, Legendre polynomial solutions of high-order linear Fredholm integrodifferential equations, J. Appl. Math. Comput. 210 (2009), 334–349.
[9] M. El- kady and M. Biomy, Efficient Legendre pseudospectral method for solving integral and Integro-differential Equations, J. Commun. Nonlinear. Sci. Numer. Simul. 15 ( 2010), 1724–1739.
[10] S. Bazm, Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations, J. Comp. Appl. Math. 275 (2015), 44–60.
[11] B.N. Mandal and S. Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials, J. Appl. Math. Comput. 190 (2007), 1707–1716.
[12] K. Maleknejad, E. Hashemizadeh and R. Ezzati, A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation, J. Commun. Nonlinear. Sci. Numer. Simul. 16 (2011), 647–655.
[13] S. Bhattacharya and B.N. Mandal, Use of Bernstein polynomial in numerical solutions of Volterra integral equations, J. Appl. Math. Sci. 36 (2008), 1773–1787
[14] Q. Wanga, K. Wanga and S. Chenb, Least squares approximation method for the solution of Volterra– Fredholm integral equations, J. Comp. Appl. Math. 272 (2014), 141–147.
[15] A. Darijani and M. Mohseni- Moghadam, Improved polynomial approximations for the solution of nonlinear integral equations, Sci. Iran. 20 (2013), 765–770.
[16] Z. Gouyandeh, T. Allahviranloo and A. Armand, Numerical solution of nonlinear Volterra-Fredholm- Hammerstein integral equations via Tau-collocation method with convergence analysis, J. Comp. Appl. Math. 308 (2016), 435–446.
[17] F. Mirzaee and E. Hadadiyan, Numerical solution of Volterra–Fredholm integral equations via modification of hat functions, J. Appl. Math. Comput. 280 (2016), 110–123.
[18] I. Aziz and S. ul- Islam, New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comp. Appl. Math. 239 (2013), 333–345.
[19] C.H. Hsiao, Hybrid function method for solving Fredholm and Volterra integral equations of the second kind, J. Comp. Appl. Math. 230 (2009), 59–68.
[20] R. Kress, Linear integral equations, Springer-Verlag, New York, 1998.
[21] G.M. Phillips, Interpolation and Approximation by Polynomials, Springer-Verlag, New York, 2003.
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Volume 14, Issue 1
January 2023Pages 2129-2142
- Receive Date: 17 February 2020
- Accept Date: 13 October 2020