scholarly journals A novel operational matrix method based on shifted Legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations

2015 ◽  
Vol 9 (2) ◽  
pp. 93-102 ◽  
Author(s):  
W. M. Abd-Elhameed ◽  
Y. H. Youssri ◽  
E. H. Doha
Keyword(s):  
Boundary Value Problems ◽  
Matrix Method ◽  
Legendre Polynomials ◽  
Boundary Value ◽  
Operational Matrix ◽  
Second Order ◽  
Singularly Perturbed ◽  
Shifted Legendre Polynomials ◽  
Operational Matrix Method

Numerical Solution of Eighth-order Boundary Value Problems by Using Legendre Polynomials

2017 ◽  
Vol 15 (02) ◽  
pp. 1750083 ◽  
Author(s):  
Anna Napoli ◽  
Waleed M. Abd-Elhameed
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Boundary Value ◽  
Operational Matrix ◽  
Harmonic Numbers ◽  
Eighth Order ◽  
Numerical Examples ◽  
Derivatives Of

The main aim of this paper is to present and analyze a numerical algorithm for the solution of eighth-order boundary value problems. The proposed solutions are spectral and they depend on a new operational matrix of derivatives of certain shifted Legendre polynomial basis, along with the application of the collocation method. The nonzero elements of the operational matrix are expressed in terms of the well-known harmonic numbers. Numerical examples provide favorable comparisons with other existing methods and ascertain the efficiency and applicability of the proposed algorithm.

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