Section 26.1
The Heat Equation
Describing how temperature (or any concentration) diffuses over time.
1
Introduction
The Heat Equation is a parabolic PDE:
is temperature. is thermal diffusivity.
2
Heuristic Derivation
Energy Balance
Rate of change of heat in a segment = Heat flowing in - Heat flowing out.
Flux is proportional to negative gradient (Fourier's Law): .
matches the equation.
3
Worked Examples
Example 1: Steady State
As , .
Then .
.
In the long run, temperature profile is a straight line connecting the boundary temps.
Example 2: Verification
Verify solves .
.
.
LHS = RHS. It works.
Example 3: Visual Diffusion
A Gaussian spike spreads out and flattens over time.
4
Practice Quiz
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