WOLFRAM|DEMONSTRATIONS PROJECT

Gibbs Phenomenon in Laplace's Equation for Heat Transfer

​
number of terms n
1
temperature of wall T
10
This Demonstration plots the solution to Laplace's equation for a square plate,
2
∂
u(x,y)
∂
2
x
+
2
∂
u(x,y)
∂
2
y
=0
.
The solution is given by
u(x,y)=
∞
∑
n=1
A
n
sin
nπ
H
ysinh
nπ
H
(x-L)
,
where
L
and
H
are the length and height of the plate (here
L=H=1
) and
A
n
=
2
Hsinh
nπ[x-L]
H

H
∫
0
g(y)sin
nπy
H
dy
,
where
g(y)=T
.
You can vary the temperature
T
along the left edge. As you increase the number of terms
n
, observe the Gibbs phenomena at the corners and along the edge where the temperature is higher.