Near-Legendre Differential Equations
DOI:
https://doi.org/10.31642/JoKMC/2018/050303Keywords:
Near-Legendre equation, Euler form, eigen polynomial.Abstract
A differential equation of the form ((1-x^2m ) y^((k)) )^((2m-k))+λy=0,-1≤x≤1,0≤k≤2m;k,m integers is called a near-Legendre equation. We show that such an equation has infinitely many polynomial solutions corresponding to infinitely many λ. We list all of these equations for 1≤m≤2. We show, for m=1, that these solutions are 'partially' orthogonal with respect to some weight functions and show how to expand functions using these polynomials. We give few applications to partial differential equations.Downloads
References
Mary Boace, Mathematical Methods in The Physical Sciences, Third Edition,John Wiley and Sons, Inc., 2006.
W. E. Boyce and R. C. DiPrima, Ordinary Differential Equations, 9th Edition, John Wiley and Sons, Inc., 2009.
G. Calbo, L. Villafuerte, Solving the random Legendre differential equation: Mean square power series solution and its statistical functions, Computers & Mathematics with Applications, Volume 61, Issue 9, Pages, 2782-2792 DOI: https://doi.org/10.1016/j.camwa.2011.03.045
Ali Davari, Abozar Ahmadi, New Implementation of Legendre Polynomials for Solving Partial Differential Equations, Applied Mathematics, 2013, 4, 1647-1650 DOI: https://doi.org/10.4236/am.2013.412224
W. N. Everitt, R. Wellman, Legendre polynomials, Legendre-Stirling numbers, and left-definite spectral analysis of Legendre differential expression, Journal of Computational and Applied Mathematics, Volume 148, Issue 1, 1 November 2002, Pages 213-238 DOI: https://doi.org/10.1016/S0377-0427(02)00582-4
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2018 Adel A. Abdelkarim
This work is licensed under a Creative Commons Attribution 4.0 International License.
which allows users to copy, create extracts, abstracts, and new works from the Article, alter and revise the Article, and make commercial use of the Article (including reuse and/or resale of the Article by commercial entities), provided the user gives appropriate credit (with a link to the formal publication through the relevant DOI), provides a link to the license, indicates if changes were made and the licensor is not represented as endorsing the use made of the work.