Math 401: Assignment 10 (Due in the last class)
1. For the minimization problem (from the theory of phase transitions)
Z ∞
1 0
1
min
(u (x))2 + (u2 (x) − 1)2 , H = {u ∈ C 2 ((−∞, ∞)) | lim u(x) = ±1} :
x→±∞
u∈H −∞ 2
4
(a) Find the problem the minimizing function should solve.
(b) Solve this problem (hint: multiply the ODE by u0 (x) and anti-differentiate to
get a first oder separable ODE which you can solve).
2. Consider the problem
Z
(∆u)2 − 2u dx
min
u∈H
D
where
D = [0, a] × [0, b],
H = {u(x) | u =
∂u
= 0 on ∂D}.
∂n
(a) Find the problem (PDE plus BCs) that the minimizer solves.
(b) Find an approximation to the minimizer in the form of the trial function
u(x) = c0 [1 − cos(2πx1 /a)] [1 − cos(2πx2 /b)]
(notice this satisfies the BCs).
3. Consider the eigenvalue problem
(−d2 /dx2 + x2 )u = λu,
0 < x < 1,
u(0) = u(1) = 0.
Find approximations for the first eigenvalue and eigenfunction using the RayleighRitz method with trial functions v1 (x) = sin(πx) and v2 (x) = sin(2πx). Just give the
first two non-zero terms of the Taylor expansion for your approximate eigenvalue and
eigenfunction around = 0.
4. Consider the fourth-order eigenvalue problem
d4
u = λu,
dx4
0 < x < 1,
u(0) = u0 (0) = u(1) = u0 (1) = 0.
(a) Find a Rayleigh-Ritz approximation for the first eigenvalue
Z 1
Z
λ1 = min{
(v 00 (x))2 dx | v(0) = v 0 (0) = v(1) = v 0 (1) = 0,
0
1
v 2 (x)dx = 1 }
0
using the trial functions v1 (x) = 1 − cos(2πx) and v2 (x) = 1 − cos(4πx) (notice
these satisfy the BCs).
(b) Solve the eigenvalue problem, find the first eigenvalue (to reasonable precision),
and compare it to your approximation in part (a).
Mar. 26
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