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Math 442 — Homework 4 Feedback
Questions 1 and 2. Conservation of energy for the wave equation in a region D in 3-dimensions:
E(t) =^12
D^ [u
(^2) t + c (^2) ∇u · ∇u] dV E′(t) =
D^ [ututt^ +^ c
(^2) ∇u · ∇ut] dV by the product rule
D^ [ut(c
(^2) ∇ · ∇u) + c (^2) ∇u · ∇ut] dV by the Wave Equation utt = c^2 ∆u = c^2 ∇ · ∇u = c^2
D^ ∇ ·^ (ut∇u)^ dV^ by the product rule for divergence = c^2
bdy D^ (ut∇u)^ ·^ ~n dS^ by the Divergence Theorem = 0 because:(i) under Dirichlet BC, u(~x, t) = 0 for all ~x ∈ bdy D and all t implies ut(~x, t(ii) under Neumann BC,) = 0 for all ~x ∈ bdy D ∇and allu · ~n = t ;∂u ∂n = 0 for all^ ~x^ ∈^ bdy^ D^ and all^ t.
Question 3. Dissipation of energy for the diffusion equation in a region3-dimensions: assuming u D in t =^ k∆u^ in the region^ D, we have E(t) =^12
D^ u
(^2) dV
E′(t) =
D^ uut^ dV = k
D^ u(∇ · ∇u)^ dV^ by the Diffusion Equation^ ut^ =^ k∆u^ =^ k∇ · ∇u = k
D^ [∇ ·^ (u∇u)^ − ∇u^ · ∇u]^ dV^ by the product rule for divergence = k
bdy D^ (u∇u)^ ·^ ~n dS^ −^ k
D^ |∇u|
(^2) dV by the Divergence Theorem
= −k
D^ |∇u|
(^2) dV using the Dirichlet BC u = 0 or the Neumann BC ∇u · ~n = 0 ≤ 0. Thus the energy is dissipated: E′(t) ≤ 0.
Section 2.3 #6. Letsatisfies w ≤ 0 when w t == 0 and when u − v, so that x w= 0 satisfies the diffusion equation, and, `. Then w ≤ 0 for all x, t, by the Maximum Principle. That is, u ≤ v for all x, t.
