Math 418
Laplace Equation on a Square
Monday, November 22, 2010
Separation of Variables solution to Laplace's Equation on a Square
uxx C uyy = 0 , 0 ! x ! 4 , 0 ! y ! 4
u 0, y = 0 , u 4, y = 0 , 0 ! y ! 4
u x, 0 = f x , u x, 4 = y x , 0 ! x ! 4
with plots :
Define the boundary value functions on the x axis and when y = 4 .
f x d piecewise 0 ! x ! 2, 1 K x K 1 , 2 ! x ! 4, 1Kx K 3 : y x d 2 K x K 2 :
Display the values on the boundary the square.
display spacecurve x, 0, f x , x = 0 ..4, thickness = 2 , color = black ,
spacecurve x, 4, y x , x = 0 ..4, thickness = 2, color = black ,
spacecurve 0, y, 0 , y = 0 ..4, thickness = 2, color = black ,
spacecurve 4, y, 0 , y = 0 ..4, thickness = 2, color = black ,
axes = normal, view = 0 ..2, labels = x, y, u ; BdyPlot d % :
Define the fundamental frequency for the Fourier expansions, then the nth coefficients in the formal
solution.
w d p /4 :
2
an d
4
4
f x sin n w x dx assuming n T posint :
0
2
bn d
4 sinh 4 n w
4
y x sin n w x dx K an coth 4 n w assuming n T posint :
0
Now define the Nth partial sum of the formal solution.
Math 418
Laplace Equation on a Square
Monday, November 22, 2010
N
u x, y, N d
>
an cosh n w y C bn$sinh n w y $sin n$w$x
n=1
N
x, y, N /
>
an cosh n w y C bn sinh n w y
sin n w x
(1)
n=1
Here is a picture of the boundary values and the 13th partial sum of the formal solution.
display BdyPlot, plot3d u x, y, 13 , x = 0 ..4, y = 0 ..4, style = patchcontour, contours = 30
The surface is typical for a solution to Laplace's equation. No hot or cold spots, all local maximum and
minimum values are attained on the boundary.
This is also the shape of the surface obtained when the wire frame determined by the boundary values is
dipped into soapy water and pulled out to dry (assuming there is no gravity).