Abstract:
We study and discuss some topics on Lie groups and their applications. In chapter one we start with a short summary on differentiate manifold and Lie groups. In chapter two we discus and show the wide structure of the construction of the Space Theory, that related to Lie groups. In chapter three we present the method of continuity of groups, which represent the similarity of Lie groups in a simple pattern. In chapter four we give some applications of Lie groups applied to differential equations. In chapter five we investigate how to obtain Lie point symmetries of partial differential equations.
Further more Many important nonlinear PDEs admit lineaizing transformations. Which can often be found by symmetry methods. We introduced such transformations and has shown how they may constructed. Nevertheless. Our treatment is neither rigorous nor exhaustive. To find out more should consult the book of Bluman and Kumei [1] (1989). These authors also present a thoroughdiscussion of potential symmetries. With many examples and applications.
The Cauchy-Kovalevskaya theorem gives conditions for the existence of solutions to the Cauchy problem for an analytic system of PDEs. Olver [5] (1993) explains why the existence and uniqueness of solutions ensures that the linearized symmetry condition yields the most general (connected) symmetry group.
We intending to use SPDE should first become familiar with Grobner bases for polynomial nonlinear systems. The text by Cox. Little and O'Shea (1992) contains a very readable account of Buchberger's algorithm and is intended for newcomers to commutative algebra. The simplest introduction to DIFFGROB2 is by Mansfield and Clakson. Who describe various strategies for simplifying the determining equations. If we wish to know more about symmetry software should consult the paper by Herman.
