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On Solving Linear and Nonlinear Sixth-Order Two Point Boundary Value Problems Via an Elegant Harmonic Numbers Operational Matrix of Derivatives

W.M. Abd- Elhameed1,2
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

Computer Modeling in Engineering & Sciences 2014, 101(3), 159-185. https://doi.org/10.3970/cmes.2014.101.159

Abstract

This paper is concerned with developing two new algorithms for direct solutions of linear and nonlinear sixth-order two point boundary value problems. These algorithms are based on the application of the two spectral methods namely, collocation and Petrov-Galerkin methods. The suggested algorithms are completely new and they depend on introducing a novel operational matrix of derivatives which is expressed in terms of the well-known harmonic numbers. The basic idea for the suggested algorithms rely on reducing the linear or nonlinear sixth-order boundary value problem governed by its boundary conditions to a system of linear or nonlinear algebraic equations which can be efficiently solved by a suitable solver. The algorithms are supported by investigating the convergence and the error analysis of the used expansion. Some illustrative examples are considered aiming to ascertain the wide applicability, and the high efficiency of the suggested algorithms. The obtained numerical results are convincing and the proposed approximate solutions are very close to the analytical ones.

Keywords

Harmonic numbers, Legendre polynomials, Petrov-Galerkin method, collocation method, sixth-order boundary value problems

Cite This Article

Elhameed, W. A. (2014). On Solving Linear and Nonlinear Sixth-Order Two Point Boundary Value Problems Via an Elegant Harmonic Numbers Operational Matrix of Derivatives. CMES-Computer Modeling in Engineering & Sciences, 101(3), 159–185.



This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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