All Solutions of Standard Symmetric Linear Partial Differential Equations Have Classical Lie Symmetry

Abstract

It is proven that every solution of any linear partial differential equation with an independent-variable-deforming classical Lie point symmetry is invariant under someclassicalLie point symmetry. This is true for any number of independent variables and for equations of any order higher than one. Although this result makes use of the infinite-dimensional component of the Lie symmetry algebra due to linear superposition, it is shown that new similarity solutions, previously thought not to be classical, can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra. This result applies to all constant-coefficient equations and to many variable-coefficient equations such as Fokker-Planck equations. © 1999 Academic Press.