Abstract

This paper investigates the problem of symbolic representation for the general solution of a system of ordinary differential equations (ODEs) with symbolically defined constant coefficients in the case where some symbolic constants can vanish. In addition, the symbolic representation of eigenvectors for the system’s coefficient matrix is not unique. It is shown that standard procedures of computer algebra systems search for specific symbolic representations of eigenvectors while ignoring the other symbolic representations. In turn, the eigenvectors found by a computer algebra system can be inadequate for constructing numerical algorithms based on them, which is demonstrated by an example. We propose an algorithm for finding various symbolic representations of eigenvectors for symbolically defined matrices. This paper considers a particular system of ODEs obtained by investigating some solutions of Maxwell’s equations; however, the proposed algorithm can be applied to an arbitrary system with a normal matrix of coefficients.

中文翻译：

---

具有符号系数矩阵的ODEs系统一般解的特征向量的符号研究

摘要

本文研究了具有符号定义的常数系数的常微分方程组（ODE）的一般解的符号表示问题，其中某些符号常数可能会消失。另外，系统系数矩阵的特征向量的符号表示不是唯一的。结果表明，计算机代数系统的标准过程会搜索特征向量的特定符号表示，而忽略其他符号表示。反过来，由计算机代数系统找到的特征向量可能不足以基于它们构建数值算法，这通过一个示例得到了证明。我们提出了一种算法，用于为符号定义的矩阵找到特征向量的各种符号表示。本文考虑了通过研究麦克斯韦方程组的一些解而获得的特殊的ODE系统。然而，所提出的算法可以应用于具有正常系数矩阵的任意系统。