Geometric methods for the symbolic integration of differential equations

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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.