On Mapping Linear Partial Differential Equations to Constant Coefficient Equations

A constructive algorithm is developed to determine whether or not a given linear p.d.e. can be mapped into a linear p.d.e. with constant coefficients. The algorithm is based on analyzing the infinitesimals of the Lie group of transformations leaving invariant the given p.d.e. As consequences, in two dimensions, necessary and sufficient conditions are given for mapping: (1) a parabolic p.d.e. into the heat equation; (2) a hyperbolic p.d.e. into the wave equation; (3) an elliptic p.d.e. into Laplace’s equation or the Helmholtz equation. The corresponding mappings are given explicitly.