半无限区间上物理奇异二阶边值问题的前牛顿法与后牛顿法的比较
A comparison between pre-Newton and post-Newton approaches for solving a physical singular second-order boundary problem in the semi-infinite interval
摘要：
本文介绍了两种基于牛顿迭代法和谱算法的数值方法来求解牛顿-费米方程。该方程是一个边界条件为无穷远的非线性奇异常微分方程。在这些方案中，牛顿方法与谱方法相结合，其中之一，通过牛顿方法将非线性常微分方程转化为线性常微分方程序列，然后使用谱方法求解。另一种方法是利用谱方法将非线性常微分方程转化为非线性代数方程组，然后用牛顿法求解。在这两种方法中，谱方法都是基于有理Gegenbauer函数的分数阶。最后，从精度、运行时间和迭代次数等方面对两种格式的计算结果进行了比较。数值实验表明，我们的方法是准确的，迄今为止所获得的最好的结果。
英文摘要：
In this paper, two numerical approaches based on the Newton iteration method with spectral algorithms are introduced to solve the Thomas-Fermi equation. That Thomas-Fermi equation is a nonlinear singular ordinary differential equation (ODE) with boundary condition in infinite. In these schemes, the Newton method is combined with a spectral method where in one of those, by Newton method we convert nonlinear ODE to a sequence of linear ODE then, solve them using the spectral method. In another one, by the spectral method the nonlinear ODE be converted to system of nonlinear algebraic equations, then, this system is solved by Newton method. In both approaches, the spectral method is based on fractional order of rational Gegenbauer functions. Finally, the obtained results of two introduced schemes are compared to each other in accuracy, runtime and iteration number. Numerical experiments are presented showing that our methods are as accurate as the best results which obtained until now.
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