Symbolic Computation and Differential and Difference Equations

符号计算与微分和差分方程

• 批准号: 9731507
• 负责人: Michael Singer
• 金额: $ 12.66万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9731507&HistoricalAwards=false
• 关键词: Symbolic; Computation; Differential; Difference; Equations

The goal of this project is to develop algorithms to determine the algebraic structure of solutions of differential and difference equations. In particular, the project seeks to find a complete efficient algorithm to compute the Galois groups of differential equation and specific algorithms to compute properties of the equations as reflected in these groups (e.g., solvability in finite terms and solvability in terms of lower order equations). Refined criteria will be sought that will allow one to construct differential equations with a specified Galois group and extend the existing solution for connected linear algebraic groups to arbitrary linear algebraic groups. The recently developed Galois theory of difference equations will be applied to similar problems for these equations as well. In particular, effective algorithms will be developed to determine if difference equations can be solved in finite terms, greatly generalizing the work of Petkovsek, Wilf, and Zeilberger. Furthermore, algorithms will be developed to determine the Galois groups of such an equation, and a constructive solution of the inverse problem for these equations will be obtained.

该项目的目标是开发算法来确定微分方程和差分方程解的代数结构。 特别是，该项目寻求找到一种完整有效的算法来计算微分方程的伽罗瓦群，以及计算这些群中反映的方程属性的特定算法（例如，有限项的可解性和低阶方程的可解性）。 我们将寻求细化的标准，使人们能够用指定的伽罗瓦群构造微分方程，并将连通线性代数群的现有解扩展到任意线性代数群。 最近发展的差分方程伽罗瓦理论也将应用于这些方程的类似问题。 特别是，将开发有效的算法来确定差分方程是否可以用有限项求解，从而极大地推广 Petkovsek、Wilf 和 Zeilberger 的工作。 此外，将开发算法来确定此类方程的伽罗瓦群，并且将获得这些方程的反问题的构造解。