Foundations — Heat equation (parabolic) 1D — derivation from Fourier's law

This page assumes you have seen nothing. We collect every symbol the parent note parent topic uses and build each one from a picture before it is allowed to appear.

0. The stage: a rod and two rulers

Before any symbol, picture the object we study.

1. $x$ — position along the rod

The picture: a point sliding left–right along the rod. Why the topic needs it: heat flows along the rod, so we must be able to name where on the rod something happens. Without $x$ we cannot say "hotter on the right."

2. $t$ — time

The picture: a stopwatch running while the rod sits there. Why the topic needs it: temperature changes — the whole point of the heat equation is to predict the future. A quantity that changes needs a clock.

3. $u(x,t)$ — the temperature field

Why the topic needs it: $u$ is literally the unknown we solve for. The PDE is an equation about $u$.

4. The derivative — "rate of change" as a slope

Two symbols coming up ($u_t$ and $u_x$) are both derivatives, so we must build that idea first.

Why the topic needs it: heat flow is about change — change in time (does this spot warm up?) and change in space (is it hotter to the right?). Both are slopes.

5. $u_t$ and $\frac{\partial u}{\partial t}$ — change in time

The curly $\partial$ (say "partial") just means "derivative while keeping the other input frozen." Since $u$ has two inputs, we must say which one we are wiggling — that is the only reason for the new symbol instead of the plain $\mathrm{d}$.

The picture: stand at one point of the rod; watch its temperature rise or fall on a thermometer. Why the topic needs it: the left-hand side of $u_t=\alpha u_{xx}$ is this quantity. It is the answer we want: how the future differs from now.

6. $u_x$ — the temperature gradient (slope in space)

The picture: the tilt of the temperature curve in the snapshot figure. Uphill to the right means $u_x>0$ (hotter to the right).

Why the topic needs it: Fourier's law says heat flows down this slope. No slope, no flow. The gradient is what drives heat.

7. $u_{xx}$ — curvature (slope of the slope)

This is the most important symbol on the whole parent page, so we give it a figure.

Why the topic needs it: it is the entire right-hand side of the heat equation. Curvature is the engine.

8. $q(x,t)$ — heat flux

The picture: an arrow through a cross-section of the rod; its size is $q$, its direction is the sign. Why the topic needs it: heat "moving" is what the flux measures. Fourier's law links flux to the gradient; conservation links flux to storage.

9. The material constants: $\rho$, $c$, $k$, and $\alpha$

The picture: $\rho c$ = the size of a bucket at each point; $k$ = the width of the pipes between buckets; $\alpha$ = how quickly the water levels even out. Why the topic needs it: the storage term $\rho cu$ and Fourier's $k$ combine in the derivation to give the single constant $\alpha$ in front of $u_{xx}$.

10. $\int$, $\Delta x$, and $A$ — the control-slab tools

The picture: a thin coin-shaped slice of the rod; we track energy into its left face and out of its right face. Why the topic needs it: the derivation balances "energy stored in the slab" against "heat flowing across its faces," then shrinks $\Delta x\to 0$. See Conservation Laws and Continuity Equation.

11. "Parabolic" — the classification word

The picture: contrast with Wave Equation (hyperbolic) 1D (things travel and bounce) and Laplace Equation (steady-state heat) (nothing changes in time). Parabolic sits between: it relaxes toward the steady state.

Prerequisite map

Equipment checklist

Give the plain-words meaning of each before you reveal it.

Connections

• Parent: Heat Equation 1D derivation
• Fourier's Law of Conduction
• Conservation Laws and Continuity Equation
• Classification of PDEs (elliptic, parabolic, hyperbolic)
• Thermal Diffusivity and Material Properties
• Separation of Variables for the Heat Equation
• Wave Equation (hyperbolic) 1D
• Laplace Equation (steady-state heat)