CHOICE OF A BASIS TO SOLVE THE LANE-EMDEN EQUATION
Abstract
If the set of basis functions is chosen by overlooking physics of a problem, then the results can be misleading. It is shown that for the Lane-Emden equation, a set of functions with semi-infinite domain sometimes fails to produce results of desired accuracy. A qualitative analysis of the problem shows that the solution is bounded when $m$ is an odd integer but is unbounded when $m$ is even. Solution of the Lane-Emden equation with rational Legendre functions, as basis, is poorer in accuracy when $m = 2$ as compared with the one when $m = 3$ with the same basis. Since the physically important region is contained in a finite interval, a set of scaled Legendre polynomials, as basis, produces results which are much more accurate on the interval of interest.
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