Moving Frames And Differential Invariants Constructed Algorithm And Its Differential Equations In Applied Research

This thesis presents a classical algorithm and an improved recursive method to construct moving frames and differential invariants based on the moving frame theory developed by Peter J. Olver and Mark. Fels. It takes several examples to demonstrate the constructive processes of two methods respectively for a Lie transformation group. The results indicate that the recursive algorithm has more advantages than the classical Cartan approach, which can be applied to arbitrary group actions systematically. More importantly, it doesn't need the existing of a slice. Especially for multi-parameter transformation group, this recursive method is more convenient when to construct the corresponding moving frames and differential invariants. It is important that the corresponding Maurer-Cartan forms could be obtained as by-products step by step. In addition, the thesis also gives a new method based on moving frames theory for solving ordinary differential equations (ODES) and partial differential equations(PDES). The results presented here not only are new, but also provide a fundamental theoretical tool to the application study of differential equations for differential invariants.Our work focus on the following six parts:Part1briefly introduces not only the importance of differential equations in the problem of nonlinear differential equations, the wide applications and the develop-ment of moving frame, but also introduces the application of differential invariants and establish the relationship between the moving frame and differential invariants. Finally, we introduced the main work of this thesis.Part2mainly elaborates the theory of the transformation group, the prolonged of group action and jet space, which provide the basis theory for the moving frame of part3.Part3illustrate the classical cartan approach and the recursive algorithm to construct moving frame in detail and demonstrate the process of the two methods respectively with examples. Also, we introduce the knowledge of the differential invariants and the differential invariant operators and their connection. Part4primary use two transformation groups as an examples to demonstrate the process of construct moving frame and the differential invariants by the classical cartan method and the recursive algorithm, also, we maked a comparison of the two methods in order to process the efficiency of the improved recursive algorithm.Part5introduce the applications of differential invariants which constructed by moving frame in the solutions of differential equations, which includes not only ordinary differential equations, but also partial differential equations.Part6is a summary and short view of this thesis.