A new method based on evolutionary and numerical algorithms for solving different forms of the nonlinear Lane-Emden equations

Subject Areas : Numeric Analyze

Seyed Reza Mirshafaei ¹ , Hashem Saberi Najafi ² , Esmaeel Khaleghi ³ , Amir Hosein Refahi Sheikhani ⁴

1 - Department of Applied Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.
2 - Department of Applied Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.
3 - Department of Mechanical Engineering, Faculty of Engineering, University of Guilan, Rasht, Iran
4 - Department of Applied Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.

Received: 2022-08-29 Accepted : 2022-11-14 Published : 2024-05-20

Keywords: معادلات دیفرانسیل لین-امدن, ساختار ستاره ای, روش های عددی, برنامه سازی ژنتیکی,

Abstract :

Evolutionary algorithms such as neural networks, genetic algorithms, and genetic programming are used in various engineering fields with the advancement of computers and the development of the processor manufacturing industry. But in mathematics, fewer than such methods have been used to solve special problems such as differential equations. This study aims to present a new and innovative method in applied mathematics and astrophysics, according to which mathematical methods can be combined with evolutionary processes, and it can be used to solve Lane-Emden nonlinear second-order differential equations. In this study, a novel GPNLE method based on an evolutionary algorithm (including genetic programming) and combining it with a numerical method (Runge-Kutta) is presented to generate mathematical models with appropriate accuracy from the solution of the second-order nonlinear Lane-Emden differential equations arising from astronomy. The present method's accuracy, efficiency, and flexibility based on numerical experiments performed on these equations have been analyzed and compared with a powerful method based on Chebyshev polynomials. The obtained result confirms the efficiency and validity of the presented method.

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