Geometric methods for the symbolic integration of differential equations

该提案的主要目标是开发新的方法，精确解常微分方程和偏微分方程。在这个建议中所包含的研究课题的动机是最近的进展，在微分方程的几何分析领域的符号方法和最新的理论发展。 理论的发展提供了一个新的，统一的，连贯的方法，以解决技术的安全性。在对称计算的符号方法和在微分几何和李理论的计算符号软件的实施进展提供了测试这些理论建设的实际效用的计算环境。 拟议的研究活动平衡数学形式主义和严谨性，希望创造有效和高效的新算法，用于解决微分方程，建立在现有的解决方案的方法和补充。几乎所有的过程中，物理科学和工程是由一种或另一种微分方程建模。 通过研究这些微分方程及其解，人们可以理解这些过程是如何演变的。 数学家使用分析、数值、几何和代数方法来研究微分方程。这一建议涉及几何和代数方法，因为这些方法最适合在交互式计算机代数系统（CAS）中实现。这一工作将扩大CAS可以求解的方程的数量和种类。 这项提案的一个独特之处是研究人员能够迅速实施新的理论进展，并将其分发给数学家，科学家和工程师的广泛客户。此外，研究人员开发的计算机软件对于培训高年级本科生和研究生几何和代数方面的重要课题非常有用。

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.