M312 #002 Sp17
Partial Differential Equations – Homework #9
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Homework #9 – PDEs in Higher Dimensions
1. Solve the IBVP: 𝑢𝑡 = 𝑢𝑥𝑥 + 𝑢𝑦𝑦 , 0 < 𝑥 < 𝜋, 0 < 𝑦 < 𝜋, 0 < 𝑡 < ∞ with boundary
conditions 𝑢(0, 𝑦, 𝑡) = 𝑢(𝜋, 𝑦, 𝑡) = 0, 𝑢𝑦 (𝑥, 0, 𝑡) = 𝑢𝑦 (𝑥, 𝜋, 𝑡) = 0, and initial condition
𝑢(𝑥, 𝑦, 0) = sin 𝑥 + 5 sin(2𝑥) cos(𝑦). Then compute the asymptotic steady-state.
2. A square plate of length and height 𝜋 meters has its surface area insulated. Its left and
right sides are also insulated while its top and bottom sides are in water baths at 0°𝐶.
Assuming the lower left point of the plate is at position (𝑥, 𝑦) = (0,0), the plate has an
initial temperature profile of 𝑓(𝑥, 𝑦) = sin 𝑦 + cos(𝑥) sin(2𝑦). Setup and solve the PDE
to compute the temperature of the plate 𝑢(𝑥, 𝑦, 𝑡) at any position 𝑥, 𝑦 and any time 𝑡.
Then use your solution to compute the temperature profile of the plate after a long time
has passed. For simplicity, assume a diffusivity of 1 m2/s in the heat equation.
3. Solve the IBVP: 𝑢𝑡𝑡 = 𝑢𝑥𝑥 + 𝑢𝑦𝑦 , 0 < 𝑥 < 𝜋, 0 < 𝑦 < 𝜋, 0 < 𝑡 < ∞, with boundary
conditions 𝑢𝑥 (0, 𝑦, 𝑡) = 𝑢𝑥 (𝜋, 𝑦, 𝑡) = 0, 𝑢𝑦 (𝑥, 0, 𝑡) = 𝑢𝑦 (𝑥, 𝜋, 𝑡) = 0, and initial
conditions 𝑢(𝑥, 𝑦, 0) = 1 + cos 𝑥 and 𝑢𝑡 (𝑥, 𝑦, 0) = 2 + 3 cos(𝑥) cos(𝑦).
4. A vibrating square membrane of length and height 𝜋 meters has all four edges fixed at
position 𝑢 = 0. Assuming the lower left point of the membrane is at position (𝑥, 𝑦) =
(0,0), it has an initial displacement given by the formula 𝑓(𝑥, 𝑦) = 3 sin 𝑥 sin(2𝑦) +
4 sin(3𝑥) sin(5𝑦) with no initial velocity. Setup and solve the PDE to compute the
position of the membrane 𝑢(𝑥, 𝑦, 𝑡) at any point 𝑥, 𝑦 and any time 𝑡. For simplicity,
assume a value of 𝑐 = 1 in the wave equation.
5. Solve the BVP: 𝑢𝑥𝑥 + 𝑢𝑦𝑦 + 𝑢𝑧𝑧 = 0, 0 < 𝑥 < 𝜋, 0 < 𝑦 < 𝜋, 0 < 𝑧 < 𝜋, with boundary
conditions 𝑢(0, 𝑦, 𝑧) = 𝑢(𝜋, 𝑦, 𝑧) = 0, 𝑢(𝑥, 0, 𝑧) = 𝑢(𝑥, 𝜋, 𝑧) = 0, 𝑢(𝑥, 𝑦, 𝜋) = 0 and
𝑢(𝑥, 𝑦, 0) = sin(𝑥) sin(2𝑦).
6. A metal cube has a length, width, and height of 𝜋 meters. Assuming a corner of the cube
is at position (𝑥, 𝑦, 𝑧) = (0,0,0), the four sides of the cube are insulated, its bottom side is
in a water bath at 0°𝐶, and its top side has a temperature profile given by the formula
𝑓(𝑥, 𝑦) = 1 + 2 cos 𝑥 + 3 cos(𝑥) cos(2𝑦). Setup and solve the PDE to compute the
steady-state solution for the temperature of the cube at any position (𝑥, 𝑦, 𝑧).