Linear 1+1 Parabolic Equations: A Complete Invariant Characterization

Abstract: 

A complete invariant characterization of scalar linear (1+1) parabolic equations under equivalence transformations for all the four canonical forms are determined. Firstly semi-invariants under changes of independent and dependent variables and the construction of the relevant transformations that relate two parabolic equations are given. Then necessary and sufficient conditions for a (1+1) parabolic equation, in terms of the coefficients of the equation, to be reducible via equivalence transformations to the one-dimensional heat equation and the three canonical equations are presented. These invariant conditions provide practical criteria for reduction to the respective canonical equations. Also the construction of the transformation formulas that yield the reductions are provided. Ample examples are given to illustrate the many results.