Differential Equations › Section I · Partial differential equations › Topic I.5
Solving the heat equation
Separation of variables gives building blocks; superposition and Fourier series assemble them into a solution that matches any reasonable starting temperature. This topic completes the method for a bar with cold ends, a bar with insulated ends and a thin ring.
In this topic
- Three steps to a full solution
- Cold ends: the product solutions
- Superposition and the initial condition
- Worked example: a starting profile made of sines
- Worked example: a bar at a uniform temperature
- How the solution behaves
- Insulated ends
- Worked example: an insulated bar evening out
- A thin ring
- Common mistakes to avoid
- Key takeaways
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