Complete Symmetry Groups: A Connection Between Some Ordinary Differential Equations and Partial Differential Equations (original) (raw)

The concept of complete symmetry groups has been known for some time in applications to ordinary differential equations. In this Thesis we apply this concept to partial differential equations. For any 1+1 linear evolution equation of Lie’s type (Lie S (1881) U¨ ber die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichung Archiv f ¨ ur Mathematik og Naturvidenskab 6 328-368 Translation into English by Ibragimov NH in CRC Handbook of Lie Group Analysis of Differential Equations 2 73-508) containing three and five exceptional point symmetries and a nonlinear equation admitting a finite number of Lie point symmetries, the representation of the complete symmetry group has been found to be a six-dimensional algebra isomorphic to sl(2,R) �s A3,1, where the second subalgebra is commonly known as the Heisenberg-Weyl algebra. More generally the number of symmetries required to specify any partial differential equations has been found to equal the numb...