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June 2018 Harmonic numbers operational matrix for solving fifth-order two point boundary value problems
Y. H. Youssri, W. M. Abd-Elhameed
Tbilisi Math. J. 11(2): 17-33 (June 2018). DOI: 10.32513/tbilisi/1529460019


The principal purpose of this paper is to present and implement two numerical algorithms for solving linear and nonlinear fifth-order two point boundary value problems. These algorithms are developed via establishing a new Galerkin operational matrix of derivatives. The nonzero elements of the derived operational matrix are expressed explicitly in terms of the well-known harmonic numbers. The key idea for the two proposed numerical algorithms is based on converting the linear or nonlinear fifth-order two BVPs into systems of linear or nonlinear algebraic equations by employing Petrov-Galerkin or collocation spectral methods. Numerical tests are presented aiming to ascertain the high efficiency and accuracy of the two proposed algorithms.


The authors would like to thank the referees for carefully reading the paper and also for their constructive and valuable comments which have greatly improved the paper.


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Y. H. Youssri. W. M. Abd-Elhameed. "Harmonic numbers operational matrix for solving fifth-order two point boundary value problems." Tbilisi Math. J. 11 (2) 17 - 33, June 2018.


Received: 15 September 2017; Accepted: 23 January 2018; Published: June 2018
First available in Project Euclid: 20 June 2018

zbMATH: 07172261
MathSciNet: MR3954180
Digital Object Identifier: 10.32513/tbilisi/1529460019

Primary: 65M70
Secondary: 35C10 , 42C10 , 65N35

Keywords: fifth-order BVPs , Galerkin and collocation methods , harmonic numbers , Legendre polynomials , Operational matrix

Rights: Copyright © 2018 Tbilisi Centre for Mathematical Sciences


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Vol.11 • No. 2 • June 2018
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