A New Algorithm And Its Application Moving Frame In Partial Differential Equation

This thesis proposed a improved new algorithm to construct moving frames and the corresponding differential invariants by research the new bonding point between the equivalent moving frames and the traditional Lie groups theory,which base on the equivalent moving frames theory.This algorithm relies on classical theoretical innovation and integration of the advantage of the computer algebra and symbolic computation system, using Maple software to avoid complex manual higher order differential calculation. The universal applicability of the new algorithm in the finite dimensional Lie groups and infinite dimensional Lie pseudo-groups action been verified by a classical Lie pseudo-groups as an example. At the same time, this thesis further explore the relationship between the moving frames theory and the exact solution of partial differential systems,group foliation which based on the moving frames theory is used as an effective way to study the exact solution of differential equation, and gives specific symbolic computation methods and procedures,finally use this method to get the variable separation solutions of group foliation of the1+1dimensional nonlinear partial differential equation.The specific work of this thesis is as follows:Chapter1briefly introduces three main development course of moving frames, recent status and results of domestic and foreign research,at the same time, the main work of this thesis is introduced.Chapter2elaborates the preparatory knowledge of the the moving frames by introducing the basic concepts and related theorems of manifold, transformation groups, group action, vector field,which laid a solid theoretical foundation for better study the moving frames and contribute to the understanding of the third and fourth chapter contents.Chapter3mainly explains the core idea of the equivalent moving frames and algebraic structure process, differential invariants and joint invariant theory, while some classic transformation groups as an example, so that we have a basic under-standing of moving frames theory. Chapter4first introduces Maurer-Cartan form,moving coframes and pseudo-groups, then focuses on the new algorithm for moving frame, taking a classic Lie pseudo-groups as an example to verify the effectiveness and simplicity of the algo-rithm.Chapter5discusses syzygy which is the relationship between differential in-variants, group foliation method of moving frames and the implementation steps of computer algebra. Last the new exact solutions of separation of variables of a1+1dimensional partial differential equations are obtained by the group foliation method.Chapter6concluded the work of the full text and proposed the next step to research questions.