The Heat Equation

Mathematical Modeling · Axiom Academy

Understanding how temperature diffuses through space and time using partial differential equations Imagine a metal rod with one end heated. What happens? Heat flows from hot regions to cold regions. The rate of heat flow depends on how steep the temperature gradient is - steeper gradients drive faster heat flow. We derive the heat equation by considering energy conservation in a small segment. The rate of temperature change equals the net heat flux into the segment, which depends on the curvature (second derivative) of the temperature profile. The constant alpha in the heat equation is called the thermal diffusivity. It combines material properties: thermal conductivity, density, and specific heat capacity. Higher diffusivity means heat spreads faster. The heat equation smooths out temperature differences over time. Sharp peaks flatten, and the temperature eventually reaches a uniform equilibrium. The characteristic time scale depends on the system size squared divided by diffusivity. To solve the heat equation, we need boundary conditions specifying what happens at the edges of our domain. Common types include fixed temperature (Dirichlet), fixed heat flux (Neumann), or insulated boundaries.

This is the written version of the interactive lesson above. See the full Mathematical Modeling course.