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Analytic and numerical solutions to the Lane-Emden equation. (English) Zbl 1223.85004
Summary: We present analytical solutions to the Lane-Emden equation $$y''(x)+\frac{2}{x}y'(x)+f(y)=0$$ describing the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules. Solutions are obtained by using the traditional power series approach and by using the Homotopy Analysis Method (HAM). We show that the series solutions obtained by the HAM converge in a larger interval than in the case of the corresponding traditional series solutions. Furthermore, we obtained numerical solutions (using the Runge-Kutta-Fehlberg 4-5 technique) which are used to validate the analytical solutions.

##### MSC:
 85A15 Galactic and stellar structure 34A34 Nonlinear ordinary differential equations and systems 65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
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