Mathematical Sciences: Closed Form Solutions of DifferentialEquations

这项数学工作将集中在微分方程解的代数结构上。这意味着分析微分方程解如何用某些特殊微分方程解的形式来表示。这些问题的具体实例包括用初等函数或Liouvillian函数隐式或显式地求解微分方程，或者更一般地，用低阶微分方程解的形式。还将对相关伽罗华群的计算进行相关调查。这项工作分为三个部分。首先，将对初等函数和其他特殊函数的代数结构进行研究，目的是改进和扩展现有的有限项积分算法。第二个重点是继续努力发展Liouvillian第一积分理论，即可用积分和指数的代数组合表示的第一积分，并开发寻找此类积分的算法。通过第一次积分，我们可以理解一个在微分方程组的解上消失的多元函数。这个项目的第三个目标是了解线性微分方程解的代数结构。在这里，人们感兴趣的是使用代数变元、李理论和伽罗瓦理论来确定给定的微分方程解是否可以表示为低阶方程的解的组合。与此相伴随的是提供用于完成所需表示的算法的基本问题。除了应用于微分方程的一般理论之外，这项研究还与计算机科学前沿的重要工作有直接关系，特别是关于符号处理算法的工作。

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.