Math 339, computation problem due Friday, Nov. 6, 2009.
Do the worked problem of Chapter 7 in our text, i.e., solve the initial boundary
value problem on [0,1]
u
t
=
2u
x2
u(0, t) =
0,
u(x, 0) =
(1)
u(1, t) +
u
(1, t) = 0
x
x
(2)
(3)
by expanding in the a
Math 339, homework problem due Monday Oct. 26, 2009.
1. Let the integral operator K act on functions g in L2 [0, 1] that vanish at the
endpoints x = 0 and x = 1 by
1
(Kg )(x) =
K (x, x )g (x )dx
(1)
0
where the kernel function K (x, x ) is the one from th
Math 339, homework problem due Wednesday Oct. 21, 2009.
1. Let the integral operator K act on functions g in L2 [0, 1] that vanish at the
endpoints x = 0 and x = 1 by
1
(Kg )(x) =
K (x, x )g (x )dx
(1)
0
where the kernel function K (x, x ) is the one from
Math 339, homework problem due Monday Oct. 19, 2009.
1. Consider the 2-point boundary value problem on the interval [0,1]
d2 f
dx2
f (0) = f (1)
= g (x)
(1)
=
(2)
0.
Verify by direct computation that the solution is
1
f (x) =
K (x, x )g (x )dx
(3)
K (x, x
Math 339, homework problem due Wednesday, Nov. 30, 2009.
1. By Sturm-Liouville theory or otherwise, show that the square integrable
solutions to
(1 x2 )f + [ ( + + 2)x]f + n(n + + + 1)f = 0
(1)
for dierent integer values of n are orthogonal with respect t
Math 339, homework problem due Wednesday, Nov. 18, 2009.
1. Consider the Laplace-Beltrami operator on the sphere (this is just the Laplacian restricted to the coordinates (, ) of the unit sphere)
2
f=
1
sin
sin
f
+
1 2f
.
sin2 2
(1)
Use the separation o
Math 339, Problem for Wednesday, Nov. 4
In spherical polar coordinates (r, , ), such that
x=
y=
z=
r sin cos
r sin sin
r cos
(1)
(2)
(3)
the gradient squared of a function f is
| f |2 =
f
r
2
+
1
r2
f
2
+
1
2 sin2
r
f
2
(4)
and the natural inner produ
Math 339, homework problem due Monday, Nov. 2, 2009.
1. Consider the dierential operator
L=
d2
d
+ A(x)
+ B (x)
2
dx
dx
(1)
on the interval [a,b], and suppose f1 and f2 are two linearly independent solutions to the homogeneous equation Lf = 0, i.e.,
Lf1 =
Math 339, nal (computational) homework problem, due by end of exam period.
Solve the heat equation
f
2f
=
t
x2
on the interval [0,1] subject to boundary conditions
f
(1, t) = 0
x
f (0, t) = 0;
and initial condition
f (x, 0) = x3 x4
(1)
(2)
(3)
First say w
Math 339, homework problem due Monday, Dec. 7, 2009.
Verify by substitution into the heat equation
f
2f
= 2 2
t
x
that
G(x, t) =
1
2 t
exp
(1)
x2
42 t
(2)
is a solution to the heat equation for all x and t > 0. Furthermore, show that
G(x, t) dx = 1
for a
Math 339, homework problem due Friday, Dec. 4, 2009.
Use the convolution formula to prove a version of the Wiener-Khinchin theorem.
Let
f (x)eikx dx ,
F (k ) =
(1)
and call S (k ) = |F (k )|2 the power spectral density. Since F (k ) is in general
complex,
Math 339, homework problem due Wednesday, Dec. 2, 2009.
Although it is not so clear how it was arrived at, we hear that
sin2 t
dt =
t2
(1)
Try to justify this statement numerically. Describe carefully in writing what
you did and what you found.
1
Math 339, computation problem due Friday, Oct. 23, 2009.
1. Consider the 2-point boundary value problem on the interval [0,1]
d2 f
2 f
dx2
f (0) = f (1)
= g (x)
(1)
=
(2)
0.
By representing both sides as a sine series, nd the Fourier coecients for the
so
Math 339, computation problem due Friday Oct. 30, 2009.
Let the integral operator K act on functions g in L2 [0, 1] that vanish at the
endpoints x = 0 and x = 1 by
1
(Kg )(x) =
K (x, x )g (x )dx
(1)
0
where the kernel function K (x, x ) is the one from th
We have a Fourier representation for the triangle function
x
0 x < 1/2
1 x 1/2 < x 1
f (x) =
(1)
namely
f (x)
=
bm sin(mx)
(2)
m=1
bm
4
=
m2 2
sin
m
2
.
(3)
The Fourier series makes sense even outside the domain of the original function
f . For values of