Symbolic Computation and Differential and Difference Equations

符号计算与微分和差分方程

• 批准号: 0634123
• 负责人: Michael Singer
• 金额: $ 29.51万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-09-15 至 2010-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0634123&HistoricalAwards=false
• 关键词: Symbolic; Computation; Differential; Difference; Equations

This research addresses foundational and computational issues concerning the algebraic behavior of systems of linear difference and differential equations and allows researchers to understand aspects of the qualitative behavior of solutions of these equations. This research develops algorithms that will be of use not only to mathematicians but engineers and scientists in many fields as well. These algorithms will be the basis of code appearing in symbolic computation computer packages used in the education as well as the day-to-day work of engineers and other scientists.In particular, this research investigates new and more efficient algorithms for the classical Picard-Vessiot theories of over determined systems of differential and difference equations, and develops new algorithms to determine elementary algebraic properties of under determined systems of linear partial differential and difference equations. It also extends the classical Picard-Vessiot theory and its algorithms to Galois theories with infinite dimensional groups, allowing applications to parameterized linear equations as well as some classes of nonlinear equations.

这项研究涉及有关线性差异和微分方程系统代数行为的基础和计算问题，并使研究人员能够理解这些方程解决方案的定性行为的方面。这项研究开发了算法，这些算法不仅可以用于数学家，而且还将用于许多领域的工程师和科学家。这些算法将是教育中使用的符号计算计算机软件包以及工程师和其他科学家的日常工作的代码的基础。尤其是，这项研究调查了古典Picard--过度确定的差异和差异方程系统的船只理论，并开发了新算法，以确定确定的线性部分差分和差异方程系统的基本代数特性。它还将经典的Picard-vessiot理论及其算法扩展到具有无限尺寸基团的Galois理论，从而使应用程序用于参数化线性方程以及某些类别的非线性方程。