Math 331, Spring 2020
Professor David Shirokoff (331-002)
Professor Michael Booty (331-004)
Homework 3
Due: Feb. 24 (331-002)
Due: Feb. 20 (331-004)
Instructions: Read each problem carefully. Show all your work for each problem. Discussion
with your peers is allowed (and encouraged), however each person must hand in their own written
solutions. You must also write at the top of the first page the names of anyone you may have
worked with.
Problem 1. (6 points) Find the radially-symmetric (i.e., independent of θ and z) equilibrium
temperature distribution ueq (r) in a cylindrical shell (e.g., a coaxial cable) with inner and outer
radii r1 = 1 and r2 = 2, heat diffusivity of κ = 1 (in some non-dimensional units), and a heat source
proportional to distance from the cylinder axis, r. Hint: as in all other equilibrium problems, set
time derivatives to zero:
∂u(r, θ)
1 ∂
∂u(r, θ)
=
r
+ 9r , on 1 < r < 2 , t > 0 ,
∂t
r ∂r
∂r
u(1, t) = T, u(2, t) = 0.
Problem 2. (5 points) Separate variables in the following PDE: substitute u(r, θ) = f (r)g(θ)
in the PDE, make sure to completely separate r from θ, and write down the two ordinary
differential equations satisfied by the functions f (r) and g(θ) (but do not solve them!). Note: an
extra algebraic step may be needed to completely separate r from θ.
∂u(r, θ)
∂ 2 u(r, θ) 1 ∂
+
tan θ
= 0.
r
∂r2
r ∂θ
∂θ
Problem 3. (7 points) Solve the following eigenvalue problem: find all eigenfunctions ψ(x) and
the corresponding eigenvalues λ; consider separately the cases λ < 0, λ = 0, and λ > 0 (for λ < 0
it may be convenient to set λ = −µ where µ > 0). Make a plot of the first two1 eigenfunctions
on the interval 0 < x < L.
d2 ψ
− 2 = λ ψ, on 0 < x < L ,
dx
dψ
ψ(0) = 0 ,
(L) = 0 .
dx
Note that the boundary conditions are of different type at the two ends 0 and L.
Problem 4. (7 points) Use separation of variables and the eigenfunctions you found in your
solution to Problem 3 to solve the following homogeneous (i.e., unforced) heat equation:
∂u(x, t)
∂ 2u
1
= κ 2 , on 0 < x < , t > 0,
∂t
∂x
2
∂u 1
u(0, t) = 0,
, t = 0,
∂x 2
u(x, 0) = 5 sin(πx) − 3 sin(3πx).
1Note:
The eigenvalues can be ordered by their size λ1 ≤ λ2 ≤ λ3 ≤ . . .. The first two eigenfunctions are the
functions ψ1 (x) and ψ2 (x) that correspond to the smallest two values λ1 , λ2 .