此篇論文的主要工作是介紹當我們給定一個微分方程時，我們將利用 Lie symmetry 這套工具來簡化所給之微分方程。而我們將以三個重要的例子來說明如何找到合適的 Lie symmetry。

In this work a general systematic method based on the point symmetry theory of Lie group to find an invariant transformation of a given differential equation will be introduced. A proper transformation can reduce the order, simplify the complexity, or even find the exactly solution of differential equations. Hence to find a good transformation is crucial. Three typical important examples will be illustrated how to find a suitable transformation.

Contents
1 Introduction    1
2 Preliminary    1
2.1  Lie group of transformations and Invariant solutions . . . . . . 2
2.2  Infinitesimal transformations and Infnitesimal generators . . . 3
2.3  Extension (Prolongation) in Differential Equations . . . . . . . . 7
2.3.1  Extension (Prolongation) in Ordinary Differential Equa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2  Extension (Prolongation) in Partial Differential Equa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4  The multiparameter (r-parameters) of Lie groups of transfor-
mations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 The Classical Similarity Method    20
4 Examples    21
4.1  Second-order linear homogeneous equation(Invariance under
Scaling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2  The Kdv equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3  Euler equation of ideal gasdynamics . . . . . . . . . . . . . . . . . . .25
5 Appendix    28
Reference    30

[1] W. F. Ames, R. L. Anderson, V. A. Dorodnitsyn, E. V. Ferapontov,
R. K. Gazizov, N. H. Ibragimov, and S. R. Svirshchevski¸³. CRC handbook
of Lie group analysis of differential equations. Vol. 1. CRC Press, Boca
Raton, FL, 1994. Symmetries, exact solutions and conservation laws.
[2] Gerd Baumann. Symmetry analysis of differential equations with
Mathematica®. Springer-Verlag-TELOS, New York, 2000. With 1 CD-
ROM (Windows, Macintosh and UNIX).
[3] G. W. Bluman and J. D. Cole. Similarity methods for differential equa-
tions. Springer-Verlag, New York, 1974. Applied Mathematical Sciences,
Vol. 13.
[4] George W. Bluman and Stephen C. Anco. Symmetry and integration
methods for differential equations, volume 154 of Applied Mathematical
Sciences. Springer-Verlag, New York, 2002. 
[5] George W. Bluman and Sukeyuki Kumei. Symmetries and differential
equations, volume 81 of Applied Mathematical Sciences. Springer-Verlag,
New York, 1989.
[6] B. Champagne, W. Hereman, and P. Winternitz. The computer calcu-
lation of Lie point symmetries of large systems of differential equations.
Comput. Phys. Comm., 66(2-3):319-340, 1991.
[7] Lawrence Dresner. Similarity solutions of nonlinear partial differential
equations, volume 88 of Research Notes in Mathematics. Pitman (Ad-
vanced Publishing Program), Boston, MA, 1983.
[8] James M. Hill. Differential equations and group methods for scientists
and engineers. CRC Press, Boca Raton, FL, 1992.
[9] Peter E. Hydon. Symmetry methods for differential equations. Cam-
bridge Texts in Applied Mathematics. Cambridge University Press,
Cambridge, 2000. A beginner's guide.
[10] Peter E. Hydon. Symmetry analysis of initial-value problems. J. Math.
Anal. Appl., 309(1):103-116, 2005.
[11] Shunji Kawamoto. The BÄacklund and the Galilei invariant transfor-
mations constructed by similarity variables for soliton equations. J.
Nonlinear Math. Phys., 2(3-4):398-404, 1995. Symmetry in nonlinear
mathematical physics, Vol. 1 (Kiev, 1995).
[12] Sophus Lie. Uber die integration durch bestimmte integrale von einer
klasse linearer partieller differentialgleichungen. Arch. Math., pages 328-
368, 1881.
[13] Sophus Lie. Vorlesungen uber differentialgleichungen mit bekkanten in-
fnitesimalen transformationen. Teubner, Leipzig, 1891.
[14] Elizabeth L. Mansfield and Peter A. Clarkson. Applications of the differ-
ential algebra package diffgrob2 to classical symmetries of differential
equations. J. Symbolic Comput., 23(5-6):517-533, 1997.
[15] Peter J. Olver. Applications of Lie groups to differential equations, vol-
ume 107 of Graduate Texts in Mathematics. Springer-Verlag, New York,
second edition, 1993.
[16] L. V. Ovsiannikov. Group analysis of differential equations. Academic
Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982. Translated from the Russian by Y. Chapovsky, Translation edited by
William F. Ames.
[17] L. V. Ovsjannikov. Lectures on the theory of group properties of differ-
ential equations. Novosibirsk. Gos. Univ., Novosibirsk, 1966.
[18] C. Rogers and W. F. Ames. Nonlinear boundary value problems in sci-
ence and engineering, volume 183 of Mathematics in Science and Engi-
neering. Academic Press Inc., Boston, MA, 1989.
[19] F. Schwarz. Automatically determining symmetries of partial differential
equations. Computing, 34(2):91-106, 1985.
[20] V. D. Sharma and R. Radha. Exact solutions of Euler equations of ideal
gasdynamics via Lie group analysis. Z. Angew. Math. Phys., 59(6):1029-
1038, 2008.
[21] Hans Stephani. Differential equations. Cambridge University Press,
Cambridge, 1989. Their solution using symmetries.