Differential Equations, Symmetries and Groups

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Physics

Synopsis:

The purpose of the lecture is to teach students computation of symmetries of the differential equations.

Requirements:

Knowledge of multivariate calculus and elementary properties of differential equation (basic solution methods, existence and uniqueness theorems).

Syllabus of lectures:

1. Symmetries in physics and mathematics

2. Groups.

3. One-parameter subgroups, generators.

4. Group actions.

5. Local and infinitesimal group actions.

6. Point transformations.

7. Symmetries of equations.

8. Determination of infinitesimal symmetries.

9. Symmetry-based reduction of order of ODEs

10. Selfsimilar solutions of PDEs

Syllabus of tutorials:

1. Calculation of symmetries of a given ODE

2. Solution of ODE via oredr reduction

3. Calculation of symmetries of a given PDE (Heat equation, KdV equation, ...)

4. Interpretation of the symmetries

5. Determination of the Lie algebra of the symmetries

6. Construction of selfsimilar solutions.

Study Objective:

Knowledge:

Lie symmetries of differential equations

Abilities:

Computation of point symmetries of differential equations and their application in solution of given ODEs and PDEs.

Study materials:

Key references:

[1] P.J.Olver, Applications of Lie Groups to Differential Equations, Springer 2000

[2] P.E. Hydon: Symmetry Methods for Differential Equations: A Beginner's Guide (Cambridge Texts in Applied Mathematics), CUP 2000

Recommended references:

[3] N.Kh. Ibragimov: Group analysis of ordinary differential equations an the invariance principle in mathematical physics, Uspekhi Mat Nauk 47:4 (1992) 83-144 Russian Math. Surveys 47:4 (1992) 89-156

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: