Geometric methods for the symbolic integration of differential equations

本建议的主要目的是发展常微分方程和偏微分方程精确解的新方法。本提案所包含的研究课题是由符号方法的最新进展和微分方程几何分析领域的最新理论发展所驱动的。理论的发展提供了一个新的，统一的，连贯的方法，以对称为基础的解决技术。对称计算符号方法的进步以及微分几何和李论计算符号软件的实现为测试这些理论结构的实际效用提供了计算环境。拟议的研究活动平衡了数学形式主义和严谨性，并希望创建有效和高效的新算法来解决基于和补充现有解决方法的微分方程。在物理科学和工程中，几乎所有的过程都是用这样或那样的微分方程来建模的。通过研究这些微分方程及其解，人们可以了解这些过程是如何演变的。数学家使用解析、数值、几何和代数方法来研究微分方程。由于几何和代数方法最适合在交互式计算机代数系统（CAS）中实现，因此本建议涉及几何和代数方法。这项工作将扩大CAS可求解方程的数量和种类。这一提议的一个独特之处在于，研究人员能够迅速实现新的理论进展，并将它们分发给数学家、科学家和工程师等广泛的客户。此外，研究人员开发的计算机软件对于在几何和代数的重要课题上训练高级本科生和研究生非常有用。

The principle objective of this proposal is to develop new methods for the exact solution of ordinary and partial differential equations. The research topics contained in this proposal are motivated by recent advances in symbolic methods and recent theoretical developments in the field of geometric analysis of differential equations. The theoretical developments provide for a new, unified, coherent approach to symmetry-based solution techniques. Advances in symbolic methods for symmetry computations and in the implementation of symbolic software for computations in differential geometry and Lie theory provide the computational environment for testing the practical utility of these theoretical constructions. The proposed research activities balance mathematical formalism and rigor with a desire to create effective and efficient new algorithms for solving differential equations which build upon and complement existing solution methodologies.Almost all processes in the physical sciences and engineering are modeled by differential equations of one kind or another. By studying these differential equations and their solution, one gains understanding of how these processes evolve. Mathematicians use analytical, numerical, geometric and algebraic methods to study differential equations. This proposal deals with geometric and algebraic methods since these methods are best suited for implementation in interactive computer algebra systems (CAS). This work will expand the number and kinds of equations which can be solved by CAS. A unique feature of this proposal is the investigators ability to quickly implement new theoretical advances and distribute them to a very broad clientel of mathematicians, scientists and engineers. In addition, the computer software developed by the investigators is very useful for training advanced undergraduate and graduate students in important topics in geometry and algebra.