PhD Qualify Exam, PDE, Mar. 02, 2012 Show all works
E: easy, M: moderate, D: difficult, O: old exam
1.(O) Solve the wave equation for infinite vibrating string utt= c2(x)uxx, where c(x) =
( c1, x < 0.
c2, x > 0.
Let a wave u(x, t) = f (x − c1t) come in from the left. Thus the initial conditions are u(x, 0) = f (x) and ut(x, 0) = −c1f0(x). Assume that u(x, t) and ux(x, t) are continuous everywhere. Also give an
interpretation for the solution you find. [10%]
2.(O) Let u, v ∈ C1(Ω) be conjugate harmonic functions, i.e., ux = vy and uy = −vx, in a simply connected domain Ω with C1 boundary in R2. Show that on the boundary curve ∂Ω, du
dn = dv ds, dv
dn = −du
ds, where d
dn denotes differentiation in the direction of the outer normal and d
ds differentiation in the counter-clockwise tangential direction. Show that these relations can be used to reduce the Neumann
problem for u to the Dirichlet problem for v. [10%]
3.(O) Let Ω ⊂ Rn be open. Show that if there exists a function u ∈ C2(Ω) vanishing on ∂Ω for which the quotient
R
Ω|∇u|2 R
Ωu2 reaches its infimum λ, then u is an eigenfunction for the eigenvalue λ, so that
∆u + λu = 0 in Ω. Let us call them λ1 and u1. How do we find λ2 and u2? Give an example of eigenvalue problem and find lim
n→∞λn in your example. [10%]
4.(a)(O) Let u be a solution of the wave equation in all of R3 × R. Suppose that a > 0 and that u(x, 0) = ut(x, 0) = 0 for |x| ≥ a. Show that u(x, t) = 0 in the double cone |x| ≤ |t| − a for |t| ≥ a. [10%]
(b)(E) Answer the same question for the wave equation in R2× R. [5%]
(c)(M) Find the fundamental solution (or Riemann function, or Green’s function, or source function) S(x, t) for the wave equation and the solution u of
(
utt− ∆u = f (x, t), x ∈ R3, t ∈ R,
u(x, 0) = g(x), ut(x, 0) = h(x), x ∈ R3, in terms
of S, f , g, and h. [10%]
(d)(M) Use Fourier transform to find the solution u(x, t) in terms of bf (ξ, t), bg(ξ), and bh(ξ). [5%]
5. In this problem set we always assume that the Neumann function H exists.
(a)(E) Analogous to the Green’s function G, please state the definition of the Neumann function H(x, y) for the operator −∆ and the domain D ∈ R2 at the point x0 ∈ D. [5%]
(b)(M) Find the solution of the problem
( ∆u = f, in D
∂u
∂n = h, on∂D. (Hint: Use the Green’s Identities.) [5%]
(c)(D) Solve the Neumann problem in the half-plane
( ∆u = f, in {y > 0},
∂u
∂n = h, on{y = 0}, with u bounded at
∞. (Hint: Consider the problem satisfied by v = ∂u
∂y.) [10%]
1
6.(M) Find the solution for the diffusion equation on the half-line:
ut− uxx = f (x, t), u(x, 0) = g(x), u(0, t) = h(t),
x > 0, t > 0,
where g(0) = h(0) = 0. [10%]
7.(D) Find a traveling wave solution of ut+uxxx+6uux = 0 (−∞ < x < ∞), that is, u(x, t) = f (x−ct).
Also assume that f (x), f0(x), f00(x) tend to 0 as x tends to ±∞. (Hing: f (x) = 1
2c sech2 1 2
√c(x − x0),
where x0 is an integration constant and c the wave speed.) [10%]
Figure 1:
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