The connection of ordinary differential equations and partial differential equations through a common realisation of the algebra of their complete symmetry groups
The complete symmetry group of a second-order 1+1 steady-state partial differential equation has been demonstrated to be represented by the eight-dimensional Lie algebra of point symmetries for which the eight-dimensional algebra is sl(3,R). We construct a steady-state partial differential equation using the idea of complete symmetry groups from a second-order free particle ordinary differential equation y" = 0. The equation immediately realized turn out to be equivalent (in symmetry structure) to a system of second-order ordinary differential equation and a first-order ordinary differential equation. Also we construct further second-order equations from higher order equations of the form yn = 0.