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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 ›› 2016, Vol. 33 ›› Issue (5): 506-516.doi: 10.3969/j.issn.1005-3085.2016.05.006

• • 上一篇    下一篇

脉冲微分方程的一个修正block-by-block数值格式

曹俊英1,2,  马群长1,  王自强1,2   

  1. 1- 贵州民族大学理学院，贵阳 550025
    2- 北京大学数学科学学院，北京 100871
  • 收稿日期:2015-07-30 接受日期:2016-05-09 出版日期:2016-10-05 发布日期:2016-12-15
  • 基金资助:
    国家自然科学基金 (11501140; 11426074)；贵州省科学技术基金 ([2014]2098; [2013]2144).

A Modified Block-by-block Numerical Scheme for Impulsive Differential Equations

CAO Jun-ying1,2,  MA Qun-zhang1,  WANG Zi-qiang1,2   

  1. 1- College of Science, Guizhou Minzu University, Guiyang 550025
    2- School of Mathematical Sciences, Perking University, Beijing 100871
  • Received:2015-07-30 Accepted:2016-05-09 Online:2016-10-05 Published:2016-12-15
  • Supported by:
    The National Natural Science Foundation of China (11501140; 11426074); the Foundation of Guizhou Science and Technology Department ([2014]2098; [2013]2144).

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摘要： 本文利用修正的block-by-block方法针对脉冲微分方程构造了高阶数值格式．修正的block-by-block方法是传统的block-by-block方法的改进，其优点是除第一块外其余每块都能够解耦求解积分方程的高阶数值方法．首先，把脉冲微分方程等价转化为脉冲型积分方程，并利用修正的block-by-block方法进行离散，得到在两个相邻脉冲点中除第一块外其余每块都解耦的高阶数值格式．其次，利用离散的Grownwall不等式证明了数值解逼近精度为四阶．最后，一系列的数值算例验证了理论分析的正确性．

关键词: 脉冲微分方程, block-by-block算法, 高阶数值格式, 收敛性分析

Abstract: In the paper, we use a modified block-by-block method to establish a high order numerical scheme for the impulsive differential equation. The modified block-by-block method is an improvement of classical block-by-block method. It is the high order numerical method for solving integral equation, and it has the advantages of the decoupled solution form at every block except the first block. Firstly, we transform the impulsive differential equation into the impulsive integral equation. Based on the impulsive integral equation form of impulsive differential equation, we establish its high order numerical scheme via modified block-by-block method. The high order numerical scheme is a decoupled solution form at each block in two adjacent impulsive points except the first block. Secondly, using the discrete Grownwall inequality, we prove that the convergence order of the numerical solution is 4. Finally, we present a series of numerical examples to support the theoretical results.

Key words: impulsive differential equation, block-by-block method, high order numerical scheme, convergence analysis

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