The direct method of functional separation of variables can provide more exact solutions than the compatibility analysis of PDEs based on a single differential constraint

Abstract

This note shows that in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, nonlinear Klein--Gordon type equations, and hydrodynamic boundary layer equations. A few new exact solutions are given.