Math 257/316 Assignment 9
Due Wed. Apr. 1

 uxx + uyy = 0 0 < x < 2 , 0 < y < 1
ux (0, y) = 0
ux (2, y) = 1
.
1. Find the solution to the boundary value problem:

u(x, 0) = 0
u(x, 1) = 0
2. Find the solution to Laplace’s equation in the semi-infinite strip, {(x, y) | 0 < x <
2, y > 0}, with the following boundary conditions:
3πx
5πx
u(0, y) = 0, ux (2, y) = 0 u(x, 0) = sin
− 2 sin
, lim u(x, y) = 0.
y→∞
4
4
3. Consider Laplace’s equation, uxx + uyy = 0, in the unit square, {(x, y) | 0 < x <
1, 0 < y < 1}, with the following Neumann boundary conditions:
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ux (0, y) = 0, ux (1, y) = y − , uy (x, 0) = 0, uy (x, 1) = 0.
2
Determine whether or not this problem has a solution, and if it does, find the solution
(which would be unique only up to a constant!).
4. 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.
1