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Supplementary Problems

(b) What are the hottest and coldest temperatures?
◦
(c) Can you choose γ so that the temperature on its outer boundary is 20 C?
(Hint: The temperature depends only on the radius.)

Exercise 215. Solve u xx + u yy = 0 in the rectangle 0 < x < a, 0 < y < b with
the following boundary conditions: u = −a on x = 0, u = 0 on x = a, u = b
y
x
x
on y = 0, u = 0 on y = b.
y
Exercise 216. Solve the following Neumann problem in the cube (0 < x < 1,
0 < y < 1, 0 < z < 1): ∆u = 0 with u (x, y, 1) = g(x, y) and homogeneous
z
Neumann conditions on the other five faces, where g(x, y) is an arbitrary function
with zero average.

Exercise 217. Solve u xx + u yy = 0 in the disk (r < a) with the boundary
condition u = 1 + 3 sin θ on r = a.

3
Exercise 218. Same for the boundary condition u = sin θ. (Hint: Use the
3
identity sin θ = 3 sin θ − 4 sin 3θ.)

Exercise 219. Solve u xx + u yy = 0 in the exterior (r > a) of a disk, with the
boundary condition u = 1 + 3 sin θ on r = a, and the condition at infinity that u

be bounded as r → ∞.

Exercise 220. Solve u +u yy = 0 in the disk r < a with the boundary condition
xx
∂u − hu = f(θ), where f(θ) is an arbitrary function. Write the answer in terms of
∂r
the Fourier coefficients of f(θ).

Exercise 221.
(a) Find the steady-state temperature distribution inside an annular plate (1 < r <
2), whose outer edge (r = 2) is insulated, and on whose inneredge (r = 1) the
2
temperature is maintained as sin θ. (Find explicitly all the coefficients, etc.)
(b) Same, except u = 0 on the outer edge.

Exercise 222. Find the harmonic function u in the semidisk (r < 1, 0 < θ < π)
with u vanishing on the diameter (θ = 0, π) and u = π sin θ − sin 2θ on r = 1.

Exercise 223. An annular plate with inner radius a and outer radius b is held at
temperature B at its outer boundary and satisfies the boundary condition ∂u/∂r =
A at its inner boundary, where A and B are constants. Findthe temperature if
it is at steady state. (Hint: It satisfies the two-dimensional Laplace equation and

depends only on r.)

Exercise 224. Solve u +u yy = 0 in the wedge r < a, 0 < θ < β with the BCs:
xx
u = θ on r = a, u = 0 on θ = 0, and u = β on θ = β. (Hint: Look for a function
independent.of r.)

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