@Article{NMTMA-4-2,
author = {},
title = {Chebyshev Spectral Methods and the Lane-Emden Problem},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2011},
volume = {4},
number = {2},
pages = {142--157},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.2011.42s.2},
url = {https://global-sci.com/article/90690/chebyshev-spectral-methods-and-the-lane-emden-problem}
}