MATH 220: PROBLEM SET 8
DUE THURSDAY, DECEMBER 6, 2012
Problem 1. Let (x) = |x| on [, ]. Let
f (x) = a0 + a1 cos x + a2 cos(2x) + b1 sin x + b2 sin(2x),
With what choice of the coecients aj and bj is the L2 error f minimal?
(Here f 2 = |f (x) (x)|2 dx.)
P
MATH 220: PROBLEM SET 7
DUE THURSDAY, NOVEMBER 29, 2012
Problem 1. Solve the inhomogeneous heat equation on the half-line for Dirichlet
boundary conditions:
ut kuxx = f, u(x, 0) = (x), u(0, t) = 0,
in two dierent ways:
(i) Using Duhamels principle, and th
MATH 220: PROBLEM SET 6
DUE THURSDAY, NOVEMBER 15, 2012
Problem 1.
(i) On R3 , nd the Fourier transform of the function g (x) = |x|1 . (Hint: to
do this eciently, consider g (x) as the limit of ga (x) = ea|x| |x|1 , and use
your result from the last probl
MATH 220: PROBLEM SET 5
DUE TUESDAY, OCTOBER 30, 2012
Problem 1. Suppose that f is (piecewise) continuous on Rn with |x|N f (x) bounded
for some N > n (or indeed simply that f L1 (Rn ). Throughout this problem,
a Rn .
(i) Let fa (x) = f (x a). Show that (
MATH 220: PROBLEM SET 4
DUE THURSDAY, OCTOBER 25, 2012
Problem 1. Solve the wave equation on the line:
utt c2 uxx = 0, u(x, 0) = (x), ut (x, 0) = (x),
with
0,
1 + x,
( x ) =
1 x,
0,
x < 1,
1 < x < 0,
0 < x < 1,
x > 1.
and
0, x < 1,
2, 1 < x < 1,
(x) =
MATH 220: PROBLEM SET 3
DUE THURSDAY, OCTOBER 18, 2012
Problem 1. Let C (R) be given by
0,
1 + x,
(x) =
1 x,
0,
x < 1,
1 < x < 0,
0 < x < 1,
x > 1,
so 0, (x) = 0 if |x| 1 and R (x) dx = 1. Let j (x) = j (jx), so j (x) = 0
if |x| 1/j , and j (x) dx = 1
MATH 220: PROBLEM SET 2
DUE THURSDAY, OCTOBER 11, 2012
Problem 1. Show that the only solution u D (R) of u = 0 is u = c, c a constant
function.
Hint: u = 0 means that u( ) = 0 for all Cc (R). You need to show that
there is a constant c such that u( ) = c
MATH 220: PROBLEM SET 1
DUE THURSDAY, OCTOBER 4, 2012
Problem 1. Classify the following PDEs by degree of non-linearity (linear, semilinear, quasilinear, fully nonlinear):
(1) (cos x)ux + uy = u2 .
(2) uutt = uxx .
(3) ux ex uy = cos x.
(4) utt uxx + eu u
MATH 220: Problem Set 8
Solutions
Problem 1. By proposition 0.11 in the Inner product spaces handout, we
know that f is minimal if we take the generalized Fourier coecients.
Meaning, if we call cn (x) = cos(nx) and sn (x) = sin(nx):
a = , c0 = 1
0
(x)dx
MATH 220: Problem Set 7
Solutions
Problem 1. Lets solve the inhomogeneous heat equation on the half-line for
Dirichlet boundary conditions:
ut kuxx = f,
x 0,
u(x, 0) = (x),
u(0, t) = 0.
(1)
(i) Using Duhamels principle:
We know from the class that the sol
MATH 220: Problem Set 6
Solutions
Problem 1. g is obviously not a Schwartz function, but it is a tempered
distribution. Therefore we can make sense of the Fourier transform of the distribution g (and see it as a function since the distribution is going to
MATH 220: Problem Set 5
Solutions
Problem 1. Let f L1 (Rn ) and a Rn . The whole problem is a matter of
change of variables with integrals.
(i)
(F fa ) ( ) =
Rn
eix fa (x)dx =
Rn
eix f (x a)dx
ei(a+y) f (y )dy = eia
=
=e
Rn
ia
(F fa ) ( ).
Rn
eiy f (y )dy
MATH 220: Problem Set 4
Solutions
Problem 1.
Lets solve the wave equation on the line:
utt c2 uxx = 0,
u(x, 0) = (x),
ut (x, 0) = (x),
(1)
with
0,
1 + x,
( x ) =
1 x,
0,
x < 1,
1 < x < 0,
0 < x < 1,
x > 1,
(2)
and
0, x < 1,
2, 1 < x < 1,
(x) =
0, x >
MATH 220: Problem Set 3
Solutions
Problem 1.
Let C (R) be given by:
0,
1 + x,
(x) =
1 x,
0,
x < 1,
1 < x < 0,
0 < x < 1,
x > 1,
(1)
so that it veries 0, (x) = 0 if |x| 1 and R (x)dx = 1.
Consider (j )j 1 constructed as j (x) = j (jx), so that j (x) = 0
MATH 220: Problem Set 2
Solutions
Problem 1. Show that the only solution u D (R) of u = 0 is u = c, c a
constant function.
u = 0 in D (R) means that u( ) = 0, Cc (R). And we want to show
that:
c R, Cc (R), u( ) =
c (x)dx.
(1)
R
Cc (R)
Lets choose a 0
suc
MATH 220: PROBLEM SET 1, SOLUTIONS
DUE THURSDAY, OCTOBER 4, 2012
Problem 1. Classify the following PDEs by degree of non-linearity (linear, semilinear, quasilinear, fully nonlinear):
(1) (cos x) ux + uy = u2 .
(2) u utt = uxx .
(3) ux ex uy = cos x.
(4) u