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Homework 7

Math 442: Intro to Partial Differential Equations

(Exercises are taken from Partial Differential Equations An Introduction, Second Edition by Walter A. Strauss.)

1. Exercise §9.2 #1. Prove that ∆(u) = (∆u) for any function; that is, the laplacian of the average is the average of the laplacian. (Hint: Write ∆u in spherical coordinates and show that the angular terms have zero average on spheres centered at the origin.)

2. Exercise §9.2 #4. Solve the wave equation in three dimensions with the initial data φ ≡ 0, ψ(x, y, z) = x 2 + y 2 + z 2 . (Hint: Use (5).)

3. Exercise §9.2 #7.

(a) Solve the wave equation in three dimensions for t > 0 with the initial conditions φ(x) = A for |x| < ρ, φ(x) = 0 for |x| > ρ, and ψ|x| ≡ 0, where A is a constant. (This is somewhat like the plucked string.)(Hint: Differentiate the solution in Exercise 6(b).)

(b) Sketch the regions in space-time that illustrate your answer. Where does the solution have jump discontinuities?

(c) Let |x0| < ρ. Ride the wave along a light ray emanating from (x0, 0). That is, look at u(x0 +tv, t) where |v| = c. Prove that

t · u(x0 + tv, t) converges as t −→ ∞