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Chebyshev Spectral Methods and the Lane-Emden Problem

Chebyshev Spectral Methods and the Lane-Emden Problem
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Year:    2011

Numerical Mathematics: Theory, Methods and Applications, Vol. 4 (2011), Iss. 2 : pp. 142–157

Abstract

The three-dimensional spherical polytropic Lane-Emden problem is $y_{rr}+(2/r) y_{r} + y^{m}=0, y(0)=1, y_{r}(0)=0$ where $m \in [0, 5]$ is a constant parameter. The domain is $r \in [0, \xi]$ where $\xi$ is the first root of $y(r)$. We recast this as a nonlinear eigenproblem, with three boundary conditions and $\xi$ as the eigenvalue allowing imposition of the extra boundary condition, by making the change of coordinate $x \equiv r/\xi$: $y_{xx}+(2/x) y_{x}+ \xi^{2} y^{m}=0, y(0)=1, y_{x}(0)=0,$ $y(1)=0$. We find that a Newton-Kantorovich iteration always converges from an $m$-independent starting point $y^{(0)}(x)=\cos([\pi/2] x), \xi^{(0)}=3$. We apply a Chebyshev pseudospectral method to discretize $x$. The Lane-Emden equation has branch point singularities at the endpoint $x=1$ whenever $m$ is not an integer; we show that the Chebyshev coefficients are $a_{n} \sim constant/n^{2m+5}$ as $n \rightarrow \infty$. However, a Chebyshev truncation of $N=100$ always gives at least ten decimal places of accuracy — much more accuracy when $m$ is an integer. The numerical algorithm is so simple that the complete code (in Maple) is given as a one page table.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/nmtma.2011.42s.2

Numerical Mathematics: Theory, Methods and Applications, Vol. 4 (2011), Iss. 2 : pp. 142–157

Published online:    2011-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    16

Keywords:    Lane-Emden Chebyshev polynomial pseudospectral.

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