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## MATH 583-TOPICS IN MATHEMATICAL PHYSICS I

### (Application of Lie Groups to Differential Equations)

## Fall 1998, Spring 2003, Fall 2011

** Text Book **:
Peter J. Olver , ** Applications of Lie Groups
in Differential Equations ** , Springer-Verlag New York , 1986 (First
Edition)

** Other Books **:
1. Peter J. Olver , ** Equivalence, Invariants and Symmetry **, Cambridge
University Press, 1995.

2. Giuseppe Gaeta, ** Nonlinear Symmetries and Nonlinear Equations **,
Kluver Academic Publishers, Dordreht ,1994.

3. W.I. Fushchich , W.M. Shtelen and N.I. Serov, ** Symmetry Analysis
and Exact Solutions of Equations of Nonlinear Mathematical Physics **,
Kluver Academic Publishers, Dordreht 1989.

4. G.W. Bluman and S. Kumei , ** Symmetries and Differential Equations **,
Springer, New York, 1989.

** Contents of the course ** :
First Semester is mainly on the
symmetries of (ordinary and partial) differential equations. We shall cover
the first four chapters of the Olver's book.

** Course Schedule **
Monday 13.40-16.30 SAZ01

** Exams and Homeworks **
First Midterm Exam (%25)
Second Midterm Exam (%25)
Final Exam (%25)
Homework and Projects (%25)

** Read the The following sections in Chapter 1.
**

Lie Groups, Vector fileds and Lie Algebras

** Subjects Covered **

** February 3 **
Local transformation groups
Vector fields
Lie Algebras

** February 17 **

Symmetries of algebraic equations

** February 24 **

Groups and differential equations

** March 3 **

Prolongation of the group action

Prolongation of the vector fields

** March 10 **

** Course Syllabus**

01. Sept. 07 - Symmetries of Algebraic Equations

02. Sept. 14 - Groups and Differential Equations

03. Sept. 21 - Prolongation

04. Sept. 28 - Calculation of Symmetry Groups

05. Oct. 05 - Integration of ODEs

06. Oct. 12 - Nondegeneracy Conditions for DEs

07. Oct. 19 - Group Invariant Solutions

08. Oct. 26 - Group Invariant Solutions

09. Nov. 02 - Group Invariant Solutions

10. Nov. 09 - Symmetry Groups and Conservation Laws

11. Nov. 16 - Symmetry Groups and Conservation Laws

12. Nov. 23 - Symmetry Groups and Conservation Laws

13. Nov. 30 - Variational Calculus and symmetries

14. Dec. 07 - Variational Calculus and symmetries

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