Abstract:

Sophus Lie’s theory of extended groups applied to differential equations is one of the more successful methods of analysis of differential equations. The method requires the invariance of a system of equations under a family of transformations. We will provide an overview of the method indicating how it can be used to solve ordinary differential equations as well as find group invariant solutions to ordinary and partial differential equations. In our exposition of the method, we will indicate how we have extended the applicability of the original approach to explain new phenomena observed in the reduction of differential equations. In particular, we will touch on the concept of hidden symmetries. These are symmetries that arise ‘unexpectedly’ in the reduction and increase of order of ordinary differential equations. We will also show how these symmetries arise in partial differential equations and the impact this analysis has on some ‘well-known’ results.

 

Speaker: 

Prof. K. S. Govinder, School of Mathematics, Statistics and Computer Science University of KwaZulu–Natal, Durban, South Africa