摘要
本文主要讨论了非线性微分方程的数值求解方法。首先介绍了数值方法的分类和选择准则，以及常用的数值求解方法概述。其次，对常规迭代方法和高效稳定的数值算法进行了比较与评估，包括算法的收敛性和精度分析。最后，提出了现有数值算法的改进与优化方案，包括改进迭代法的收敛速度和稳定性、优化剖分算法和网格生成方法，以及基于机器学习的非线性微分方程求解算法改进。
关键词：非线性微分方程 数值求解方法 迭代法
Abstract
This paper mainly discusses the numerical solution of nonlinear differential equations. Firstly, the classification and selection criteria of numerical methods are introduced, and the common numerical solving methods are summarized. Secondly, the conventional iterative method and the efficient and stable numerical algorithm are compared and evaluated, including the convergence and precision analysis of the algorithm. Finally, the improvement and optimization scheme of the existing numerical algorithms are proposed, including improving the convergence speed and stability of the iterative method, optimizing the segmentation algorithm and the grid generation method, and improving the nonlinear differential equation solving algorithm based on machine learning.
Keywords: Nonlinear differential equation Numerical solution method Iterative method
目 录
1 引言 1
2 非线性微分方程的数值求解方法概述 1
2.1 数值方法的分类和选择准则 1
2.2 常用的数值求解方法概述 2
3 数值算法的比较与评估 2
3.1 常规迭代方法的性能比较 2
3.2 高效稳定的数值算法的比较与评估 3
3.3 算法的收敛性和精度分析 4
4 现有数值算法的改进与优化 4
4.1 改进迭代法的收敛速度和稳定性 4
4.2 优化剖分算法和网格生成方法 5
4.3 基于机器学习的非线性微分方程求解算法改进 5
5 结论 6
致 谢 7
参考文献 8
