# 臺灣博碩士論文加值系統

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 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 12 Bluman’s Method for the Conservation Laws with Conservation Law Multipliers 32.1 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Euler Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 The Direct Method of Construction of Conservation Laws . . . . 52.3.1 The Conservation Laws of the Harry Dym Equation . . . 62.3.2 The Conservation Laws of the Modified Harry Dym Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Invertible Mappings of Nonlinear PED system to Linear PDE System Through Conservation Law Multipliers 93.1 The Invertible Linearization Mapping . . . . . . . . . . . . . . . 93.2 Linear Operators and Adjoint Linear Operators . . . . . . . . . . 103.3 The Multipliers of the Conservation Laws for Linearizing Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.1 Linearization of an Ad Hoc Constructed Equation . . . . . 133.3.2 Linearization of the Hunter-Saxton Equation . . . . . . . 133.3.3 Linearization of the Monge Amp`ere Equation . . . . . . . 154 A Maple Program for Finding Fluxes of Conservation Laws 174.1 The Flux Equation and Line Integral Formula . . . . . . . . . . . 174.1.1 The Fluxes of the Conservation law of the KdV Equation . 184.2 The Program Process and Examples . . . . . . . . . . . . . . . . 214.2.1 The Fluxes of the Conservation Law of the KdV Equation 224.2.2 The Fluxes of the Conservation Law of the Nonlinear Telegraph Equation . . . . . . . . . . . . . . . . . . . . . . . 224.2.3 The Fluxes of the Conservation Law of the Modified Harry Dym Equation . . . . . . . . . . . . . . . . . . . . . . . 235 Conclusion 25A The Maple Code of JGH fluxes 26
 Bibliography[1] G.W. Bluman, Direct Construction Method for Conservation Laws of Partial Differential Equations. Part I: Examples of Conservation Law cassifications. Eur. J. Appl. Math. 13, 545-566, 2002.[2] G.W. Bluman, Direct Construction Method for Conservation Laws of Partial Differential Equations. Part II: General Treatment. Law cassifications. Eur. J. Appl. Math. 13, 567-585, 2002.[3] A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations. Comput. Phys. Commun. 176, 48-61, 2007.[4] A. F. Cheviakov, GeM symbolic software package (version 0.31+) for general sysmmetry and conservation law analysis of differential equations. University of Saskatchewan, Saskatoon, S7N 5E6 Canada, website: http://math.usask.ca/ cheviakov/gem/[5] A. T. S.kWan, Finding Conservation Laws for Partial Differential Equations. M.A.Sc., The University of British Columbia, 2007.[6] G. W. Bluman, Alexei F. Cheviakov, and Stephen C. Anco. Applications of Symmetry Methods to Partial Differential Equations, vol 168 of Applied Mathematical Sciences. Springer New York, 2010.[7] Alexei G. Kushner, A Contact Linearization Problem for Monge-Amp`ere Equations and Laplace Invariants. Springer Science+Business Media B.V. 2008.[8] S. C. Anco and G. Bluman, Direct Construction of Conservation Laws from Field Equations, Phys. Rev. Lett. 78, 2869-2873 (1997).[9] P. Olver, Applications of Lie Groups to Differential Equations, volume 107 of Graduate Texts in Mathematics, second edition. Springer New York, second edition, 1993.[10] T.Wolf, Partial and Complete Linearization of PDEs Based on Conservation Laws. In: Wang, D., Zheng, Z. (eds.) Trends in Mathematics: Differential Equations with Symbolic Computation, pp. 291- 306. Birkhauser, Basel, 2005.[11] Ryuzo Sato and Rama V. Ramachandran, Conservation laws and symmetry : applications to economics and finance. boston, MA : Kluwer Academic Publishers, 1990.[12] Michiel Hazewinkel, Encyclopaedia of Mathematics. Springer-Verlag Berlin Heidelberg New York 2002.[13] R. A. Saxton, Dynamic instability of the liquid crystal director. In Current Progress in Hyperbolic Systems: Riemann Problems and Computations (W. B. Lindquist, ed.), Contemporary Mathematics, vol. 100, American Mathematical Society, Providence, 1989, pp. 325-330.[14] R. M. Miura, Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation. J. Math. Phys. 9, 1202-1204, 1968.
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