Gibbs Phenomenon in Laplace's Equation for Heat Transfer
Gibbs Phenomenon in Laplace's Equation for Heat Transfer
This Demonstration plots the solution to Laplace's equation for a square plate, u(x,y)+u(x,y)=0.
2
∂
∂
2
x
2
∂
∂
2
y
The solution is given by
u(x,y)=sinysinh(x-L)
∞
∑
n=1
A
n
nπ
H
nπ
H
where and are the length and height of the plate (here ) and
L
H
L=H=1
A
n
2
Hsinh
nπ[x-L]
H
H
∫
0
nπy
H
where .
g(y)=T
You can vary the temperature along the left edge. As you increase the number of terms , observe the Gibbs phenomena at the corners and along the edge where the temperature is higher.
T
n