A New Numerical Solution to the Lane- Emden Equations with Cylindrical and Planar Polytropes Using the Monte Carlo Method

El- Essawy, Samah H; Soliman, Ahmed A.; Nouh, Mohamed I; Abdel- Rahman, Helal I; Abd- Elmougod, Gamal A

doi:10.21608/sjsci.2024.283603.1197

A New Numerical Solution to the Lane- Emden Equations with Cylindrical and Planar Polytropes Using the Monte Carlo Method

Document Type : Regular Articles

Authors

¹ Astronomy Department, National Research Institute of Astronomy and Geophysics, 11421 Helwan, Cairo, Egypt

² Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt

³ Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, 22511, Egypt

10.21608/sjsci.2024.283603.1197

Abstract

Lane-Emden equations are important because they explain a star's internal structure, interstellar matter, molecular clouds, and even spiral arms, which can be computed and utilised to forecast different physical properties. The primary goal of this research is to develop a numerical solution for the Lane-Emden (LE) type equations using the Monte Carlo (MC) method. The MC method is increasingly becoming a valuable tool in various scientific fields and has many applications in multiple disciplines. This paper proposes an MC algorithm that can solve two key problems related to LE-type equations: a 1-D problem for a planar polytrope and a 2-D problem for a cylindrical polytrope. To gauge the performance and effectiveness of the numerical solutions generated by the proposed MC algorithm, we compared them with other numerical methods mentioned in previous literature. Our findings indicate that the MC solutions align with Runge-Kutta (RK) solutions and other numerical methods used to solve these problems.

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