臺灣博碩士論文加值系統

在這篇碩士論文中，我們研究一個具有點對稱之不可交換的二維李代數之二階微分方程。從李代數的一個理想跟非理想著手，我們可以分別使用李點對稱根非局部對稱的方法使得二階微分方程降階成可積式(quadrature)，接著我們舉一些例子來說明如何使用這兩種方法並比較及對照兩者在應用上的不同。

Contents
Abstract(English) i
Abstract(Chinese) ii
Acknowledgements iii
Contents iv
1 Introduction 1
2 Review of Lie Point Symmetry Analysis 3
2.1 One-parameter transformation groups . . . . . . . . . . . . . . . . . 3
2.2 Prolongation formulae . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 The determining equations for Lie point symmetries . . . . . . . . . . 6
2.4 The method of canonical variables . . . . . . . . . . . . . . . . . . . . 8
2.5 The method of differential invariants . . . . . . . . . . . . . . . . . . 8
2.6 Lie algebras and solvable Lie algebras . . . . . . . . . . . . . . . . . . 10
2.7 The Lie point symmetry method for reduction of order for secondorder
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Non-local symmetries 14
3.1 The non-local symmetry method . . . . . . . . . . . . . . . . . . . . 14
3.2 Application of non-local symmetry method to the double reduction
of order of second-order differential equations . . . . . . . . . . . . . 16
3.3 Discussion about the Lie point symmetry method and non-local symmetry
method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
iv
4 Summary and conclusions 26
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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