A Three-Fold Approach to the Heat Equation: Data, Modeling, Numerics

Abstract

This article describes our modeling approach to teaching the one-dimensional heat (diffusion) equation in a one-semester undergraduate partial differential equations course. We constructed the apparatus for a demonstration of heat diffusion through a long, thin metal rod with prescribed temperatures at each end. The students observed the physical phenomenon, collected temperature data along the rod, then referenced the demonstration for purposes in and out of the classroom. Here, we discuss the experimental setup, how the demonstration informed practices in the classroom and a project based on the collected data, including analytical and computational components.

Keywords:

• Mathematical modeling
• partial differential equations
• student project

ACKNOWLEDGEMENTS

We are grateful for the many discussions about this experiment with Professor Sharon Stephenson in the Physics department at Gettysburg College. We thank Gettysburg College students Madison Hill and Tessa Thorsen for their help in the lab.

FUNDING

We would like to thank the Johnson Center for Creative Teaching and Learning at Gettysburg College for funding this experiment through a Johnson Teaching Grant.

Additional information

Notes on contributors

Kimberly Spayd

Kim Spayd has a B.S. in mathematics (with a concentration in applied math) from Notre Dame, M.S. in statistics from the University of North Carolina - Chapel Hill, and Ph.D. in mathematics from North Carolina State University where she began work in PDE modeling of two-phase flow in porous media. She is an assistant professor of mathematics at Gettysburg College.

James Puckett

James Puckett is an assistant professor of physics at Gettysburg College. He earned a B.S. and Ph.D. in physics at North Carolina State University. His research interests are in experimental soft matter physics, specifically granular materials and collective animal behavior.