The well-known Lane–Emden equation plays an important role in describing some phenomena in mathematical physics and astrophysics. Recently, a new type of this equation with fractional order derivative in the Caputo sense has been introduced. In this paper, two computational schemes based on collocation method with operational matrices of orthonormal Bernstein polynomials are presented to obtain numerical approximate solutions of singular Lane–Emden equations of fractional order. Four illustrative examples are implemented in order to verify the efficiency and demonstrate solution accuracy.

中文翻译：

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天体物理学中奇异分数式 Lane-Emden 型方程的数值研究

著名的 Lane-Emden 方程在描述数学物理和天体物理学中的一些现象方面起着重要作用。最近，已经引入了一种新型的具有卡普托意义上的分数阶导数的方程。在本文中，提出了两种基于搭配方法和正交伯恩斯坦多项式运算矩阵的计算方案，以获得分数阶奇异Lane-Emden方程的数值近似解。实施了四个说明性示例，以验证效率并证明解决方案的准确性。