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 some classical Lie 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. Q 1999

Nonclassical symmetries of a class of nonlinear partial differential equations with arbitrary order and compatibility

“…Through (b 3 ) and (b 4 ), we can obtain 6 ) into other equations we can obtain the determining equations for the nonclassical symmetries of the Boussinesq equation, and it reads as follows:…”

“…Recently in Ref. [6] Broadbridge and Arrigo have shown that all solutions of standard symmetric linear partial differential equations have classical Lie symmetry. In addition to the nonclassical method, the determining equations are usually highly nonlinear unlike the determining equations for the classical method which are linear (see Ref.…”

Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems

“…In what follows, we omit the cases where (11) are linear or linearizable via a point transformation, as it is known that all solutions of linear PDEs can be obtained via classical Lie symmetries [30]. Each equation will be considered separately.…”

Umbral calculus, difference equations and the discrete Schrödinger equation

“…(2.27). Thus we put 45) and must prove that the coefficients B a are finite and depend on σ in the proper way (i.e. B a = B a σ n+1−a , where B a is finite and does not depend on σ).…”