Unveiling the physics of partial differential equations with heuristics

TL;DR

This paper emphasizes the importance of heuristic and order of magnitude estimates in understanding the physics of partial differential equations, especially for students, highlighting their limitations and educational value.

Contribution

It introduces heuristic methods for analyzing PDEs and discusses their limitations, aiming to improve physics education for beginners.

Findings

Heuristic estimates reveal key features of PDEs.

Limitations of simple heuristics are discussed.

Educational benefits for physics students are highlighted.

Abstract

Heuristic arguments and order of magnitude estimates for partial differential equations highlight essential features of the physics they describe. We present order of magnitude estimates, and their limitations, for the three classic second order PDEs of mathematical physics (wave, heat, and Laplace equations), for first order transport equations, and for two non-linear wave equations. It is beneficial to expose the beginning student to these considerations before jumping into more rigorous mathematics. Yet these simple arguments are missing from physics textbooks.