Legendre Polynomials’ Second Derivative Tau Method for Solving Lane-Emden and Ricatti Equations

Authors

M. Abdelhakem, Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt\\ Basic Science Department, School of Engineering, Canadian International College, New Cairo, Egypt\\Helwan School of Numerical Analysis in Egypt (HSNAE)Follow
Mona Fawzy, Basic Science Department, October High Institute for Engineering & Technology (OHI), 6 October, Egypt\\ Helwan School of Numerical Analysis in Egypt (HSNAE)Follow
M. El-Kady, Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt\\ Basic Science Department, School of Engineering, Canadian International College, New Cairo, Egypt\\ Helwan School of Numerical Analysis in Egypt (HSNAE)Follow
Hanaa Moussa, Basic Science Department, School of Engineering, Canadian International College, New Cairo, Egypt\\ Helwan School of Numerical Analysis in Egypt (HSNAE)Follow

Author Country (or Countries)

Egypt

Abstract

This paper investigates a highly efficient method that depends on the Tau method for solving initial and boundary value problems. The second derivatives of Legendre polynomials (SDLPs) have been used as novel basis functions. A linearization relation for the presented basis functions has been introduced and proved to avoid any issues arising during tau’s integration, especially for the nonlinear problems. Consequently, some essential integrations have been determined. Moreover, we used those relations to construct explicit forms for approximating the solutions of Lane-Emden and the Recatti equations. In addition, the presented strategy’s converge and error analysis are discussed carefully and in-depth. Finally, the mentioned IVPs have been solved via the proposed method. The results have been compared with the others’ methods, which showed our technique’s accuracy, efficiency, and stability.

Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/amis/170305

Recommended Citation

Abdelhakem, M.; Fawzy, Mona; El-Kady, M.; and Moussa, Hanaa (2023) "Legendre Polynomials’ Second Derivative Tau Method for Solving Lane-Emden and Ricatti Equations," Applied Mathematics & Information Sciences: Vol. 17: Iss. 3, Article 5.
DOI: http://dx.doi.org/10.18576/amis/170305
Available at: https://digitalcommons.aaru.edu.jo/amis/vol17/iss3/5