Section 26.7
Laplace's Equation
What does the heat distribution look like when it stops changing?
1
Introduction
In 2D, the steady state Heat Equation () becomes:
Solutions are called Harmonic Functions. They have the Mean Value Property.
2
Rectangular Domain
Solution Form
Assume . We get and .
One direction will have sines/cosines (to fit 0 BCs), the other will have or exponentials.
3
Worked Examples
Example 1: Top Edge Hot
Square plate. Left, Right, Bottom = 0. Top = .
X-direction has 0 BCs .
Y-direction has 0 at bottom .
.
Example 2: Symmetry
What if on top is ?
Only the term survives.
.
Find by setting .
Example 3: Visual Field
Isotherms curve away from the hot edge.
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Practice Quiz
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