齐次线性常微分方程组基于过渡矩阵的求解方法  

Solving Method of Homogeneous Linear Ordinary Differential Equations Based on Transition Matrix

作　　者：冯俊娥[1] 张庆乐 FENG Jun-e;ZHANG Qing-le(School of Mathematics,Shandong University,Jinan 250100, China)

机构地区：[1]山东大学数学学院,济南250100

出　　处：《大学数学》2020年第6期55-62,共8页College Mathematics

基　　金：国家自然科学基金(61773371,61877036);山东省自然科学基金(ZR2019MF002)。

摘　　要：研究了齐次线性常微分方程组基于过渡矩阵的求解方法.文章首先讨论了将一个矩阵A化为约当标准型J的过渡矩阵P的代数性质,给出了过渡矩阵P的列向量与矩阵A的特征向量以及广义特征向量之间的关系,在此基础上给出过渡矩阵P的具体求法,并研究了齐次常系数微分方程组基解矩阵的列向量与过渡矩阵P的列向量之间的关系,同时给出了基解矩阵的具体构造方法.最后给出数值例子说明本文方法的有效性.This paper considers homogeneous linear ordinary differential equations based on fundamental solution matrix method.Firstly,algebraic properties are discussed for the transition matrix P,via which Jordan normal form J of the coefficient matrix A can be derived.Detailed relationships are obtained among the column vectors of the transition matrix P and character vectors and generalized character vectors of the coefficient matrix A.Based on it,constructing method of the transition matrix P is proposed.Furthermore,the relationship of column vectors are investigated between the fundamental solution matrix and the transition matrix.Simultaneously,one solving method of the fundamental solution matrix is presented.Finally,numerical examples are given to show the effectiveness of the proposed method.

关 键 词：约当标准型 过渡矩阵 基解矩阵 齐次线性常微分方程组 

分 类 号：O231[理学—运筹学与控制论]