臺灣博碩士論文加值系統

在這篇論文中，我們使用 Bluman的方法建構Harry Dym方程和modifed Harry Dym方程的守恆定律。我們也使用Bluman的方法分別找到Hunter-Saxton方程和Monge Apère方程的線性轉換。
所有計算都是使用Maple來執行。特別是，我們使用GeM封包來執行 Bluman的方法。GeM是一個Maple封包，包含四個方法尋找守恆律的fluxes。我們設計一個Maple程式來執行第五個方法，這方法是根據在flux方程和線積分上。

In this thesis, we employ Bluman’s direct method to construct conservation laws of the Harry Dym equation and the modifed Harry Dym equation. Linearizing transformations(invertible point or contact transformations) for the Hunter-Saxton equation and the Monge Ap`ere equation are found, respectively, by using the conservation law multiplier method also introduced by Bluman.
All calculations are implemented in Maple. Particularly, we use the GeM package to carry out Bluman’s method. It is a Maple package that contains four methods for finding the fluxes of the conservation laws. A Maple program is designed to realize the fifth method which is an algorithm based on flux equations and line integral formulas.

1 Introduction 1
2 Bluman’s Method for the Conservation Laws with Conservation Law Multipliers 3
2.1 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Euler Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 The Direct Method of Construction of Conservation Laws . . . . 5
2.3.1 The Conservation Laws of the Harry Dym Equation . . . 6
2.3.2 The Conservation Laws of the Modified Harry Dym Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Invertible Mappings of Nonlinear PED system to Linear PDE System Through Conservation Law Multipliers 9
3.1 The Invertible Linearization Mapping . . . . . . . . . . . . . . . 9
3.2 Linear Operators and Adjoint Linear Operators . . . . . . . . . . 10
3.3 The Multipliers of the Conservation Laws for Linearizing Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3.1 Linearization of an Ad Hoc Constructed Equation . . . . . 13
3.3.2 Linearization of the Hunter-Saxton Equation . . . . . . . 13
3.3.3 Linearization of the Monge Amp`ere Equation . . . . . . . 15
4 A Maple Program for Finding Fluxes of Conservation Laws 17
4.1 The Flux Equation and Line Integral Formula . . . . . . . . . . . 17
4.1.1 The Fluxes of the Conservation law of the KdV Equation . 18
4.2 The Program Process and Examples . . . . . . . . . . . . . . . . 21
4.2.1 The Fluxes of the Conservation Law of the KdV Equation 22
4.2.2 The Fluxes of the Conservation Law of the Nonlinear Telegraph Equation . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.3 The Fluxes of the Conservation Law of the Modified Harry Dym Equation . . . . . . . . . . . . . . . . . . . . . . . 23
5 Conclusion 25
A The Maple Code of JGH fluxes 26

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