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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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