MATH 655 Supplementary Homework
Fall 2012
1. Derive the four point property
u(x, t) + u(x + h − k, t + h + k) = u(x + h, t + h) + u(x − k, t + k)
if u is a solution of the wave equation in a domain Ω ⊂ R2 containing the tilted rectangle
with vertices at (x, t), (x + h − k, t + h + k), (x + h, t + h), (x − k, t + k). Prove the
converse statement that if the four point property holds for every such rectangle in Ω
then utt − uxx = 0 must hold. (Suggestion: Look at Taylor polynomials for u.)
2. Show that the solution w(x, t) of the Cauchy problem for the Klein-Gordon equation
wtt − wxx + w = 0
w(x, 0) = 0 wt (x, 0) = h(x)
can be expressed as
1
w(x, t) =
2
Z
x+t
p
J0 ( t2 − (x − y)2 )h(y) dy
x−t
where J0 is the zero order Bessel function defined, for example, by
2
J0 (z) =
π
Z
π
2
cos (z sin θ) dθ
0
(Suggestion: ’Descend’ from the two dimensional wave equation satisfied by u(x, y, t) =
w(x, t) cos y. By choosing h = δ, we obtain a fundamental solution of the Klein-Gordon
equation.)
3. Formulate and prove a uniqueness theorem for
utt − ∆u = f (x, t)
(x, t) ∈ ΩT
∂u
+ α(x)u = j(x, t)
(x, t) ∈ ST
∂ν
u(x, 0) = g(x) ut (x, 0) = h(x)
x∈Ω
where α(x) ≥ 0 and Ω ⊂ RN is a bounded domain with smooth boundary.