1
Underground temperature oscillations
Let u(x, t) describe the temperature at time t [sec] at a depth x [m] underground. It is governed by the partial differential equation
∂u
∂ 2u
= k 2,
∂t
∂x
where k [m2 /sec] is the thermal diffusivity. Assume the (locally averaged)
temperature overground is oscillating with a period of 1 year with the global
average, for simplicity, equal to 0,
u(0, t) = A0 cos ωt.
1. Find ω [Hz] so that the period is 1 year.
2. Let us look for the solution u(x, t) that is also periodic with period 1
year and has the structure
U (x, t) = V (x) cos ωt + W (x) sin ωt.
Find U (x, t).
(Hint: Let X(x) = V (x) − iW (x), then U (x, t) = Re (x(x)eiωt ). One
can look for X(x) and then take the real part.)
3. Let k = 5 × 10−9 [m2 /sec]. At what depth is the amplitude of temperature oscillations halved? At which depth are the seasons reversed?
1
2
Rocket propulsion
Let m = m(t) be the mass of a rocket, c be the speed at which the burnt fuel
is ejected (with respect to the rocket), v = v(t) is the velocity of the rocket.
1. Using the Newton’s Law (in the momentum form, the mass is changing!), derive the rocket propulsion equation
m
dv
dm
+c
= F,
dt
dt
2. Let F = −mg − kv, where k is the air drag coefficient. Assume that
the fuel is burned at a constant rate,
dm
= −β.
dt
Find velocity at time t (assume the rocket hasn’t run out of fuel yet)
if v(0) = v0 and m(0) = m0 .
2
3
Green’s Function for the 2nd order linear
ODE
For the equation
ax00 + bx0 + cx = g(t)
x(0) = x0 ,
x0 (0) = x1 ,
1. express the solution in terms of the weight function w(t) which is defined by
1
L(w) = 2
.
as + bs + c
(Use the Laplace Transform technique and the convolution integral.)
2. Find all possible forms of the function w(t) (depending on a, b and c).
3. In the case a = 1, b = 0, c = 4, x0 = x1 = 0 derive the same answer
using variation of parameters.
3
4
Modeling the effect of an earthquake on a
multi-storey building
We will model the building as stack of blocks (floors) connected to each other
by springs. Let xj (t) be the displacement of the j-th floor with respect to the
equilibrium position. Elastic force exerted on the block j by its displacement
with respect to the block j −1 is then given by −k(xj −xj−1 ). We will assume
all blocks have mass m = 10, 000 [kg] and the elastic constant k = 100, 000
[kg/sec2 ]. The connection between the 1st floor and the ground is governed
by the same constant.
1. Turn the equation for 7-storey building as a 7 × 7 system and find the
frequencies of the natural vibrations of the building.
2. In an earthquake the ground is moving under the first floor with the
displacement given by g(t) = E cos ωt. Find and plot the amplitude of
the movement of the top floor with respect to the driving frequency of
the earthquake ω.
4
5
Bessel functions of the first kind
The Bessel equation is
x2 y 00 + xy 0 + (x2 − p2 )y = 0.
We will assume p is a non-negative integer, p = 0, 1, 2, . . ..
1. Find the solution to the Bessel equation in the form of series
yp (x) =
∞
X
an x n .
n=0
2. Prove that the Bessel functions yp (x) satisfy
d p
(x yp (x)) = xp yp−1 (x).
dx
5
6
Wildlife preserve
You are a manager of a newly established wildlife reserve that has foxes and
rabbits. Foxes eat rabbits and rabbits eat grass. The population of rabbits
R(t) and foxes F (t) is governed by the system (the time is measured in
months)
(
dR
= 0.07R − 0.0001RF,
dt
dF
= −0.08F + 0.0003RF.
dt
On January 1st you release the initial number of F0 foxes and R0 rabbits.
1. Determine numerically the period of oscillation of the rabbit and fox
populations,
2. Find numerically the max and min of F (t) and R(t) and the dates at
which the extrema occur,
3. (*) Prove that the motion is periodic (hint: find a conserved quantity).
6