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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (6): 941-967.doi: 10.3969/j.issn.1005-3085.2023.06.007

• • 上一篇    下一篇

改进PM算法数值求解常微分方程边值问题

谢正荣1,2,  艾轶博1,  张卫冬1   

  1. 1. 北京科技大学国家材料服役安全科学中心，北京 100083
    2. 华东师范大学数学科学学院，上海 200241
  • 收稿日期:2021-05-31 接受日期:2022-06-24 出版日期:2023-12-15 发布日期:2024-02-15
  • 通讯作者: 张卫冬 E-mail: zwdpaper@163.com

Numerical Solution of Boundary Value Problems of Ordinary Differential Equations Based on Improved PM Algorithm

XIE Zhengrong1,2,  AI Yibo1,  ZHANG Weidong1   

  1. 1. National Center for Materials Service Safety, University of Science and Technology Beijing, Beijing 100083
    2. School of Mathematical Sciences, East China Normal University, Shanghai 200241
  • Received:2021-05-31 Accepted:2022-06-24 Online:2023-12-15 Published:2024-02-15
  • Contact: W. Zhang. E-mail address: zwdpaper@163.com

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摘要：

PM算法是一种常微分方程初值问题非传统数值解法。为将该算法推广至两点边值问题，作出了如下改进：首先，提出“分离原理”将常微分方程组对应的多目标优化问题转化为多个相对独立的单目标优化问题，进而为引入边界误差训练提供理论基础；随后，在一阶常微分方程组的形式下，发现了二阶常微分方程I/II型边值问题计算格式的等价性；最后，构造了“保持导数关系假设初值条件”和“引入独立参数逼近初值条件”两种技术方案，完成了对I/II/III型和混合型边值问题的数值求解。

关键词: 常微分方程, 梯度下降法, 复化梯形求积公式, 多目标优化, I/II/III型边值问题, 混合型边值问题

Abstract:

PM algorithm, an unconventional numerical method for initial value problems of ordinary differential equations, is investigated. In order to extend the algorithm to the two-point boundary value problem, the following improvements are made in this paper: Firstly, the separation principle is proposed to transform the multi-objective optimization problem corresponding to ordinary differential equations into several relatively independent single objective optimization problems; Then in the framework of the first order ordinary differential equations, the equivalence of the computational schemes for the I/II type boundary value problems of the second order ordinary differential equation is established; By maintaining the derivative relation and assuming the initial value condition or introducing independent parameters to approximate the initial value conditions, the improved PM algorithms are derived and can solve the boundary value problems of I/II/III types and even mixed type, with the second-order convergence speed.

Key words: ordinary differential equations, gradient descent method, complex trapezoid quadrature formula, multi-objective optimization, I/II/III boundary value problems, mixed boundary value problems

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