MA330: Assignment 8
Required Reading.
• Read Chapter 5, § 50-53
To be turned in May 10th at the start of class.
1. Textbook, page 135, #2
2. Textbook, page 135, #4. Hint: Define f = u∇v.
3. Textbook, page 141, #1
4. Textbook, page 142, #4
5. Textbook, page 142, #7b
6. Suppose that u and v solve the same heat conduction problem in a volume V . There is a source/sink of heat energy
inside the volume, the initial temperature distribution is known, and the heat flux at the surface is also known.
ut = ∆u + s, with ut=0 = f and n̂ · ∇u = g on the surface
vt = ∆v + s, with vt=0 = f and n̂ · ∇v = g on the surface
(a) Define w = u − v. Show that w solves the following heat conduction problem.
wt = ∆w, with wt=0 = 0 and n̂ · ∇w = 0 on the surface
(b) Following the example in class, apply the divergence theorem to the integrand w∆w to show the following:
ZZZ
wwt + k∇wk2 dV = 0
V
(c) Use parts a) and b) to show that
d
E(t) ≤ 0
dt
and
E(0) = 0
where E(t) is defined by
ZZZ
w2 dV.
E(t) =
V
(d) Use part c) to prove that E(t) = 0 and use this to prove that w = 0.
(e) Use part d) to prove that solutions to the heat conduction problem are unique, that is, u and v must be the
same function.
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