A reliable algorithm for the approximate solution of the nonlinear Lane‐Emden type equations arising in astrophysics
A reliable algorithm for the approximate solution of the nonlinear Lane‐Emden type equations arising in astrophysics
复制标题
天体物理学中非线性 Lane-Emden 型方程近似解的可靠算法
DOI：
10.1002/num.22237
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Devendra Kumar
中科院分区：
文献类型：
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作者：
Harendra Singh;Hari M. Srivastava;Devendra Kumar
本文提出了一种可靠的算法来获得天体物理中的非线性Lane-埃姆登型方程的近似解。该算法基于雅可比多项式的积分运算矩阵和配点法。给出了该方法的收敛性分析和数值稳定性。讨论了标准Lane埃姆登方程、白色矮星方程、等温气体球方程等Lane埃姆登型方程的几种有趣的非线性情形以及Richardson的电子流理论的数值结果。这些数值结果以表格和图形的形式示出，用于Jacobi多项式的特定情况，例如Legendre多项式（P1）、第二类Chebyshev多项式（P2）、第三类Chebyshev多项式（P3）、第四类Chebyshev多项式（P4）和Gegenbauer（或超球面）多项式（P5）。数值计算结果也与以前通过应用一些众所周知的和最近开发的数值方法得到的结果进行了比较，可以看出，我们的数值计算结果更准确。最大绝对误差和均方根误差通过使用P1、P2、P3、P4和P5计算以用于比较目的。
In this paper, we present a reliable algorithm to obtain the approximate solution of the nonlinear Lane‐Emden type equations arising in astrophysics. The suggested algorithm is based upon the operational matrix of integration for Jacobi polynomials and the collocation method. Convergence analysis and numerical stability of the suggested method are provided. Numerical results for several interesting nonlinear cases of the Lane‐Emden type equations such as the standard Lane‐Emden equation, the white‐dwarf equation, and the isothermal gas spheres equation, as well as Richardson's theory of thermionic current are discussed. These numerical results are shown in the form of tables and figures for the particular cases of Jacobi polynomials such as the Legendre polynomial (P1), the Chebyshev polynomials of the second kind (P2), the Chebyshev polynomials of the third kind (P3), the Chebyshev polynomials of the fourth kind (P4), and the Gegenbauer (or ultraspherical) polynomials (P5). Numerical results are also compared with those that were derived earlier by applying some well‐known and recently developed numerical methods and it is observed that our numerical results are more accurate. The maximum absolute errors and the root mean square errors are calculated by using P1, P2, P3, P4, and P5 for comparison purposes.