Mathematical Sciences: Closed Form Solutions of DifferentialEquations

数学科学：微分方程的闭式解

• 批准号: 8722093
• 负责人: Michael Singer
• 金额: $ 4.41万
• 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: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8722093&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Closed; Form; Solutions

This mathematical work will center on the algebraic structure of solutions of differential equations. By this one means the analysis of how solutions of differential equations can be expressed in terms of solutions of certain special differential equations. Particular instances of these problems include solving differential equations implicitly or explicitly in terms of elementary or liouvillian functions, or more generally, in terms of solutions of differential equations of lower order. Related investigations into the calculation of associated Galois groups will also be carried out. The work divides into three parts. In the first, work will be done on the algebraic structure of elementary and other special functions, with the aim of improving and extending existing algorithms for integration in finite terms. The second thrust involves a continuation of efforts to develop a theory of liouvillian first integrals - first integrals expressible in terms of algebraic combinations of integrals and exponentials, and developing algorithms for finding such integrals. By a first integral, one understands a function of several variables which vanishes on solutions of systems of differential equations. A third goal of this project is to understand the algebraic structure of solutions of linear differential equations. Here one is interested in using algebraic arguments, Lie theory and Galois theory to determine whether solutions of given differential equations can be expressed as combinations of solutions of equations of lower order. Coupled with this are fundamental questions of providing algorithms for accomplishing the desired representations. In addition to its applications to the general theory of differential equations, this research bears direct relationship with important work at the frontiers of computer science, particularly with regard to symbolic manipulation algorithms.

这项数学工作将集中在代数 微分方程解的结构 受此 一种是分析微分方程的解 方程可以用某些解来表示， 特殊微分方程 的特定实例 这些问题包括解微分方程 隐含地或明确地， liouvillian函数，或者更一般地说， 低阶微分方程的解。 相关 对伴随伽罗瓦计算的研究 小组也将进行。 这项工作分为三个部分。 第一，工作 将完成的代数结构的基本和 其他特殊职能，目的是改善和 扩展了现有的有限积分算法 届 第二个重点是继续努力 发展了Liouvillian第一积分理论 可表示为下列代数组合的积分 积分和指数，并开发算法， 找到这样的积分。 通过第一积分，我们可以理解 在解上为零的多元函数 微分方程组。 第三个目标 项目是了解解决方案的代数结构 线性微分方程 这里人们感兴趣的是 利用代数论证、李群理论和伽罗瓦理论， 确定给定微分方程的解 可以表示为方程的解的组合， 低一级的 与此同时， 提供用于实现期望的 表示。 除了它的应用到一般的理论， 微分方程，这项研究承担直接 与前沿重要工作的关系 计算机科学，特别是关于符号 操纵算法