# 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

## TL;DR

This paper demonstrates that the direct method of functional separation of variables can yield more precise solutions to nonlinear PDEs than the traditional compatibility analysis with a single differential constraint, supported by various examples.

## Contribution

It shows that the direct functional separation method can outperform the differential constraint approach in finding exact solutions for certain nonlinear PDEs.

## Key findings

- The direct method can produce more exact solutions than the compatibility analysis.
- Examples include reaction-diffusion, convection-diffusion, Klein-Gordon, and boundary layer equations.
- Several new exact solutions are provided.

## 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.