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Exam 2
Math 425
April 26, 2011
Jerry L. Kazdan
12:00 1:20
Directions This exam has three parts, Part A, short answer, has 1 problem (10 points). Part B
has 4 shorter problems (9 points each, so 36 points). Part C has 3 traditional problems (15 points
each so 45 points). Total is 91 points.
Closed book, no calculators or computers but you may use one 3 5 card with notes on both
sides.
Part A: Short Answer (1 problem, 10 points).
1. Let S and T be linear spaces and A : S T be a linear map. Say V and W are particular
solutions of the equations AV = Y1 and AW = Y2 , respectively, while Z = 0 is a solution of
the homogeneous equation AZ = 0.
Answer the following in terms of V , W , and Z.
a)
b)
c)
d)
e)
Find
Find
Find
Find
Find
some solution of
some solution of
some solution of
another solution
another solution
AX = 3Y1 .
AX = 5Y2 .
AX = 3Y1 5Y2 .
(other than Z and 0) of the homogeneous equation AX = 0.
of AX = 3Y1 5Y2 .
Part B: Short Problems (4 problems, 9 points each so 36 points)
B1. Suppose f is a function of one variable that has a continuous second derivative. Show that
for any constants a and b , the function
u(x, y ) = f (ax + by )
is a solution of the nonlinear PDE
uxx uyy u2 = 0.
xy
B2. U = (1, 1, 0, 1) and V = (1, 2, 0, 1) are orthogonal vectors in R4 .
Write the vector X = (1, 1, 1, 0) in the form X = aU + bV + W , where a, b are scalars and
W is a vector perpendicular to U and V .
B3. If u(x, y ) is a solution of the Laplace equation in the unit disk x2 + y 2 < 1 with boundary
conditions
1
for x2 + y 2 = 1, y > 0
u(x, y ) =
0
for x2 + y 2 = 1, y 0.
Compute u(0, 0).
1
B4. This problem concerns the solution of the initial-value problem for the wave equation utt =
uxx + uyy in two space variables (x, y ) R2 , with together the initial conditions
u(x, y, 0) = f (x, y ),
ut (x, y, 0) = 0.
If f (x, y ) is a 2 periodic functions of x , so f (x +2, y ) = f (x, y ) for all x , show that u(x, y, t)
is also a 2 periodic function of x .
Part C: Traditional Problems (3 problems, 15 points each so 45 points)
C1. Let R2 be a bounded region in the plane.
a) Let w(x, y, t) be a solution of the modied heat equation
wt = wxx + wyy 7wx + wy 5w
for (x, y ) and 0 < t T < . Show that the solution w cannot have a local positive
maximum or negative minimum at a point of .
Note: There are two cases, one where the maximum point accurs at a point (x, y, t)
with 0 < t < T and one at a point (x, y, T )
b) If w(x, y, 0) = sin(x + 2y ) for (x, y ) and 2 w(x, y, t) 3 for (x, y ) , t 0,
what can you conclude about the size of w(x, y, t) for (x, y ) , t 0?.
C2. In a bounded region Rn , let u(x, t) satisfy the modied heat equation
ut 2tu = u,
(1)
as well as the initial and boundary conditions
u(x, 0) = f (x),
with u(x, t) = 0 for x ,
in
t 0.
(2)
Let u(x, t) = (t)v (x, t). Show that by picking the function (t) cleverly, v satises the
standard heat equation vt = v as well as the initial and boundary conditions (2).
Remark: This generalized to ut + a(t)u = u where a(t) is any continuous function.
C3. The motion u(x, y, t) of a special drum R2 satisle the modied wave equation
utt + b(x, y, t)ut = u
for (x, y ) ,
t > 0.
(3)
with boundary condition
u(x, y, t) = 0
for (x, y ) , t 0.
Dene the energy
E (t) :=
1
2
u2 + | u|2 dx dy.
t
Assume that |b(x, y, t)| m for some constant m and all (x, y ) , t 0.
2
(4)
dE
2mE for all t 0.
dt
d 2mt
b) Deduce that
e
E (t) 0 for all t 0, and hence that
dt
a) Show that
E (t) e2mt E (0)
for all t 0.
c) If u(x, y, 0) = 0 and ut (x, y, 0) = 0 for (x, y ) , what does this say about E (t) for t 0
and hence about u(x, y, t) for t 0?
3