# Heat Conduction with Inhomogeneous Condition and Homogeneous Condition

The Homogeneous Demo demonstrates the fundamental mechanism of heat conduction: pure decay as the temperature profile flattens toward its stationary equilibrium of zero. The Inhomogeneous Demo proves that a complex physical scenario-the initial thermal shock and subsequent relaxation to a non-zero, linear temperature gradient-can be successfully decomposed. By subtracting the stationary solution $$T^{\text {st }}$$, we isolate the problem's difficulty into a standard transient decay problem ( $$u$$ ), which is precisely the type of solution demonstrated in the homogeneous case. The full solution is thus revealed to be nothing more than the linear, fixed steady-state profile, upon which a decaying, transient wave is temporarily superimposed.

### Narrated video

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### Condensed Notes

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