MATH 520 Homework
Spring 2014
51. Using the trial function
|x|2
R2
compute an upper bound for the first Dirichlet eigenvalue of −∆ in the ball B(0, R) of RN .
Compare to the exact value of λ1 in dimensions 2 and 3. (Zeros of Bessel functions can be
found, for example, in tables, or by means of a root finding routine in Matlab.)
φ(x) = 1 −
52. If Ω ⊂ Rn is a bounded open set with smooth enough boundary, find a solution of the
wave equation problem
utt − ∆u = 0
x∈Ω t>0
x ∈ ∂Ω t > 0
u(x, t) = 0
u(x, 0) = f (x)
ut (x, 0) = g(x)
in the form
∞
X
u(x, t) =
x∈Ω
cn (t)ψn (x)
n=1
where {ψn }∞
n=1 are the Dirichlet eigenfunctions of −∆ in Ω.
53. Consider the Sturm-Liouville problem
u00 + λu = 0
0<x<1
0
u (0) = u(1) = 0
It can be shown that the eigenvalues are the critical points of
R1 0 2
u (x) dx
J(u) = R01
2
0 u(x) dx
on the space H = {u ∈ H 1 (0, 1) : u(1) = 0}. Use the Rayleigh-Ritz method to estimate
the first two eigenvalues, and compare to the exact values. Choose polynomial trial functions
which resemble what the first two eigenfunctions should look like.
54. Let T be the integral operator
1
Z
|x − y|u(y) dy
T u(x) =
0
on L2 (0, 1). Show that
1
1
≤ ||T || ≤ √
3
6
(Suggestion: the lower bound can be obtained using a simple choice of trial function in the
corresponding Rayleigh quotient.)
55. The Abel integral equation is
Z
T u(x) =
0
x
u(y)
√
dy = f (x)
x−y
a first kind Volterra equation with a weakly singular kernel. Derive the explicit solution formula
Z x
1 d
f (y)
√
u(x) =
dy
π dx 0
x−y
Rx
(Suggestions: it amounts to showing that T 2 u(x) = π 0 u(y) dy. You’ll need to evaluate an
Rx
integral of the form y √z−ydz√x−z . Use the change of variable z = y cos2 θ + x sin2 (θ).)