Math 316 – Assignment 9
Due: Wednesday, April 30, 2011
1. Solve the boundary-value problem
uxx + uyy + u = 0,
0 < x < 1, 0 < y < 1,
u(0, y) = 0, u(1, y) = 0,
u(x, 0) = 0, uy (x, 1) = sin(3πx).
2. Consider Laplace’s equation with derivative boundary conditions:
uxx + uyy = 0,
0 < x < a, 0 < y < b,
ux (0, y) = 0, ux (a, y) = f (y),
uy (x, 0) = 0, uy (x, b) = 0.
a) Show that the general solution is of the form
∞
A0 !
nπx
nπy
u(x, y) =
An cosh(
+
) cos(
).
2
b
b
n=1
b) Show that if u(x, y) is a solution, then so is u(x, y) + const for any constant. Hence:
The solution is only unique up to a constant and the constant A0 cannot be determined.
"b
c) Show that the coefficients A1 , A2 , . . . can be determined provided that 0 f (y) dy = 0
(no net flux through the boundary). (Otherwise there is no solution to this problem.)
3. Find the solution of Laplace’s equation in a wedge of angle 0 < α < 2π with insulating
boundary conditions on the sides θ = 0 and θ = α:
1
1
urr + ur + 2 uθθ = 0,
0 < r < a, 0 < θ < α,
r
r
uθ (r, 0) = 0, uθ (r, α) = 0,
u(a, θ) = f (θ), u(r, θ) bounded as r → 0.
If α = 2π the same solution holds, corresponding to a full circle with an insulating barrier
along the line θ = 0. If f (θ) = cos(θ/2) in this case, write down the solution u(r, θ) and try
to sketch it as a 3D surface plot.
4. Excel spreadsheet Use separation of variables to solve the following boundary-value problem
uxx + uyy = 0,
−1 < x < 1, −1 < y < 1,
u(−1, y) = 0, u(1, y) = 0,
u(x, −1) = sin 2πx, u(x, 1) = − sin 2πx.
Then use a spreadsheet to calculate the solution numerically (you might like to use the excel
file on the web as a template), and check that it agrees with your analytical solution. Hand
in a print out of your spreadsheet with a plot of the solution. Hint: You may want to use that
u(x, 1) = −u(x, −1).