Heat Equation Steady State

Differential Equations · Axiom Academy

A metal rod, two fixed end temperatures, and the finite-difference trick that turns a differential equation into a system of linear equations you can just solve. A rod is held at 100°C on the left and 0°C on the right. Wait long enough and the temperature along it stops changing — that steady state is what we're about to compute three different ways. Set the ends, watch the middle With no heat source anywhere along the rod, the steady state satisfies the simplest possible differential equation: d²T/dx² = 0. Drag either end and see what shape that forces on everything between. Add a source — and watch a hot spot appear Now the rod generates its own heat internally (a wire carrying current, say). The equation becomes d²T/dx² = −q/k. Drag the source strength and watch the straight line bow into a curve — past a point, the hottest spot on the rod is hotter than either end . Does a finer grid actually help — here? Drag the number of grid points from a coarse 3 up to a fine 49 and watch the error between the finite-difference solve and the exact formula. For THIS problem the answer is surprising: it doesn't shrink, because it's already at the floor.

This is the written version of the interactive lesson above. See the full Differential Equations course.