Symbolic Computation and Differential and Difference Equations

符号计算与微分和差分方程

• 批准号: 0096842
• 负责人: Michael Singer
• 金额: $ 20.83万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2006-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0096842&HistoricalAwards=false
• 关键词: Symbolic; Computation; Differential; Difference; Equations

Michael F. Singer proposes research to develop efficient algorithms to determine the algebraic structure of solutions of differential and difference equations. In particular the investigator proposes to find efficient algorithms to compute the Galois groups for large classes of differential equations and work towards finding a general algorithm to calculate the Galois group of any linear differential equation. He proposes to also find efficient 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) and apply these algorithms to integrability problems of Hamiltonian systems. The investigator will also use these algorithms to give efficient methods to determine properties of algebraic equations (e.g., absolute irreducibility, calculation of Galois groups). He proposes to find refined criteria that will allow one to construct differential equations with a specified Galois group and extend his solution of the inverse problem for connected linear algebraic groups to arbitrary linear algebraic groups. He will apply his recently developed Galois theory of difference equations to similar problems for these equations as well. In particular he proposes to refine the algorithms to determine if difference equations can solved in finite terms and extend this to q-difference equations, greatly generalizing the work of Petkovsek, Wilf and Zeilberger, develop algorithms to determine the Galois group of such an equation and give a constructive solution of the inverse problem for these equations.

迈克尔·辛格（Michael F. Singer）提出的研究旨在开发有效的算法，以确定差分方程解决方案解决方案的代数结构。特别是，研究者建议找到有效的算法来计算大型微分方程的Galois组，并致力于找到一般算法来计算任何线性微分方程的Galois组。他建议还发现有效的算法来计算这些组中反映的方程属性（例如，以有限的术语和较低方程式的溶解性为单位），并将这些算法应用于汉密尔顿系统的集成性问题。研究者还将使用这些算法来提供有效的方法来确定代数方程的特性（例如，绝对不可约性，GALOIS组的计算）。他建议找到精致的标准，以允许一个人使用指定的Galois组构建微分方程，并将其针对连接的线性代数组的反问题扩展到任意线性代数组。他将把他最近开发的GALOIS差异方程式理论应用于这些方程式的类似问题。特别是他提议优化算法，以确定差异方程是否可以以有限的术语解决，并将其扩展到Q-差异方程式，从而大大推广了Petkovsek，Wilf和Zeilberger的工作，并开发了算法来确定这些方程的Galois组，并为这些方程提供反向问题的建设性解决方案。