Math 401: Assignment 1 Solutions
1. (Fun with generalized functions)
(a) Show f (x)y (x) = f (y )y (x) (where y (x) = (x y ) denotes the delta function
centred at the point y ).
Using the denition of multiplication of a distribution by a function, we have
Math 401: Assignment 5 Solutions
1. A smooth function u is called subharmonic (respectively superharmonic) if u 0
(respectively u 0).
(a) Show that if u is subharmonic (say, in R3 , for simplicity though the Rn case is
essentially the same) then
u(x) the
Math 401: Assignment 4 Solutions
1. Let G(y ; x) be the (Dirichlet) Greens function for and a region D (in R2 , say).
Prove (rigorously!) the symmetry of G: G(y ; x) = G(x; y ). (Hint: set v (z ) := Gx (z )
and w(z ) := Gy (z ), and apply Greens second id
Math 401: Assignment 3 Solutions
1. Find the free-space Greens function for the operator + k 2 (k > 0 a constant) in
R3 . That is, solve
( + k 2 )Gx = x
in R3 , with Gx (y ) 0 as |y | . (Hint: look for Gx (y ) in the form Gx (y ) = g (r)/r,
r := |y x|.)
W
Math 401: Assignment 2 Solutions
2
d
d
1. Let L := a0 (x) dx2 + a1 (x) dx + a2 (x).
(a) Show that L = L if and only if a0 = a1
We have (carry out the integration by parts if it is unclear)
L u = (a0 u) (a1 u) + a2 u = a0 u + (2a0 a1 )u + (a0 a1 + a2 )u.
S
Math 401: Assignment 6 Solutions
1. Consider the heat-equation on the half-line with insulating boundary condition at the
endpoint:
x > 0, t > 0
ut = uxx
ux (0, t) = 0
.
u(x, 0) = u0 (x), x > 0
(a) Write down the problem that the corresponding (Neumann)
Math 401: Assignment 7 Solutions
1. Wave equation on the half-line. Solve the wave equation on the half-line, with
zero initial velocity, with Dirichlet BCs at the origin
0 < x < , t > 0
utt c2 uxx = 0,
u(0, t) = 0
,
u(x, 0) = g (x), ut (x, 0) = 0
and wi
Math 401: Mid-Term Test
Feb. 11, 2011
Total: 30 points.
50 minutes. No notes allowed. Show all your work and justify all your answers.
1. Consider the following ODE problem for u(x), 1 x 2:
(x2 u ) = f (x),
u (1) = 3, u (2) = 0.
(a) (2 pts.) Write the sol
Math 401: Mid-Term Test Solutions
1. Consider the ODE BVP for u(x), 0 x :
u + u = f (x),
u(0) = 0, u (0) = u ( ).
(1)
(a) (5 pts.) Find the (homogeneous) adjoint problem.
d2
Solution: let L = dx2 +1, and supose u(x) satises the above boundary conditions.
Math 401: Assignment 10 Solutions
1. For the minimization problem (from the theory of phase transitions)
min
uH
1
1
(u (x)2 + (u2 (x) 1)2 ,
2
4
H = cfw_u C 2 (, ) |
lim u(x) = 1 :
x
(a) Find the problem the minimizing function should solve.
If u(x) is the
Math 401: Assignment 8 Solutions
1. Consider the following Neumann ODE eigenvalue problem on D = [0, 1]:
2
d
( dx2 + x) =
0<x<1
,
(0) = 0 = (1)
a small, positive number.
(a) Find the maximum and minimum values of x on D, and use this knowledge to
nd upp
Math 401: Assignment 9 Solutions
1. (Euler-Lagrange equations) Let D Rn be a bounded domain. For the following
functionals and boundary conditions, nd the Euler-Lagrange equation and boundary
conditions satised by extremizers (maximizers or minimizers):
(