Computational Mathematics and Computer Modeling with Applications (CMCMA)

Numerical solution of differential equations of Lane-Emden type by Gegenbauer and rational Gegenbauer collocation methods

https://doi.org/10.52547/CMCMA.1.1.69
Volume 1, Issue 1 - Serial Number 1
June 2022
Pages 69-85
Computational Mathematics and Computer Modeling with Applications (CMCMA)

Document Type : Regular paper

Authors

Fatemeh Baharifard 1
Kourosh Parand 2

1 School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

2 Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

Abstract
In this paper, we apply the collocation method for solving some classes of Lane-Emden type equations that are determined in interval $[0, 1]$ and semi-infinite domain. We use an orthogonal system of functions, namely Gegenbauer polynomials and introduce the shifted Gegenbauer polynomials and the rational Gegenbauer functions as basis functions in the collocation method for problems in interval $[0, 1]$ and semi-infinite domain, respectively.
We estimate that the proposed method has super-linear convergence rate
and also investigate the Gegenbauer parameter $ (\alpha)$ to get more accurate answers for various Lane-Emden type problems. The comparison between the proposed method and other numerical results shows that the method is efficient and applicable. 

Keywords

Gegenbauer polynomials
Rational Gegenbauer functions
Collocation method
Nonlinear ODE
Lane-Emden equations
Astrophysics
  • Receive Date 12 May 2023
  • Revise Date 31 May 2023
  • Accept Date 30 June 2023

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