# Polytropic Dynamical Systems with Time Singularity

• Oday Hazaimah Northern Illinois University, USA
Keywords: nonlinear Lane-Emden equation, Euler's method, shooting method, polytropes, dynamics, singularities

### Abstract

In this paper we consider a class of second order singular homogeneous differential equations called the Lane-Emden-type with time singularity in the drift coefficient. Lane-Emden equations are singular initial value problems that model phenomena in astrophysics such as stellar structure and are governed by polytropics with applications in isothermal gas spheres. A hybrid method that combines two simple methods; Euler's method and shooting method, is proposed to approximate the solution of this type of dynamic equations. We adopt the shooting method to reduce the boundary value problem, then we apply Euler's algorithm to the resulted initial value problem to get approximations for the solution of the Lane-Emden equation. Finally, numerical examples and simulation are provided to show the validity and efficiency of the proposed technique, as well as the convergence and error estimation are analyzed.

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Published
2024-05-20
How to Cite
Hazaimah, O. (2024). Polytropic Dynamical Systems with Time Singularity. Earthline Journal of Mathematical Sciences, 14(4), 721-746. https://doi.org/10.34198/ejms.14424.721746
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Articles