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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?
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An example of slow manifold
We will study the system
(
dx
dt
dy
dt
= −xy,
= −y + x2 .
(1)
1. Plot several trajectories of the system for −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1. Make sure
you plot enough to see the pattern: all trajectories go to zero, but close to zero they
stick together to flow along the same curve. This is the “center manifold” or “slow
manifold”.
2. Linearize the system, find and plot its eigen-directions. Is the zero point of nonlinear
system linearly stable, unstable or indeterminate?
3. The “slow subspace” is the eigen-direction corresponding to eigenvalue 0. Identify the
slow subspace in our problem. The “slow manifold” is the set of trajectories tangent
to the slow subspace. We will look for the slow manifold given by the series expansion
of the form
y = f (x) = ax2 + bx3 + cx4 + · · ·
(2)
There is no constant term in the expansion because we want the manifold to pass
through zero. Why is there no linear (∼ x) term?
4. To find the manifold, we remember that it’s made up of solutions to the system. Differentiate both sides of y = ax2 +bx3 +cx4 +· · · with respect to t and use equations (1) and
(2) to get an algebraic (no derivatives!) equation in the variable x only. Use matching
in powers of x to find sevaral terms of expansion of f (x). Plot the resulting function on
top of the trajectories of the system. Describe the dynamics of the trajectories. Why
is the manifold called “slow”?
5. (*) Use the method of Lyapunov function (Section 9.6 of the book) to prove that zero
is a stable point: show that V (x, y) = x2 + y 2 − x2 y is a suitable function.
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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
L(w) =
1
.
as2 + 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.
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Modeling the effect of an earthquake on a multistorey 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 14 × 14 first-order 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.
3. (*) Do the same for the intermediate floors. Notice that for some resonance frequencies,
some intermediate floors don’t move so much. Can you see it in the eigenvectors of the
matrix?
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Column/beam buckling
The structural stability of beams and columns can be investigated using the Euler-Bernoulli
equation
d4 w
d2 w
EI 4 + P 2 = 0,
(3)
dx
dx
where w(x) is the lateral displacement, E and I are constants (if the beam is uniform)
describing the physical properties of the beam and P is the pressure compressing the beam.
Ends of the beam are either “pinned”
w(x0 ) = w00 (x0 ) = 0,
(4)
w(x0 ) = w0 (x0 ) = 0.
(5)
or “fixed” (“clamped”)
Here x0 is either 0 or L, where L is the length of the beam.
There is always the “trivial” solution to the equation and boundary conditions, w(x) ≡ 0.
The beam “buckles” when the pressure P reaches the critical value at which a non-trivial
solution appears (why does uniqueness theorem not apply here?). We will find the critical
values for different combinations of conditions.
1. Rewrite equation (3) in the form
2
d4 w
2d w
+
k
= 0,
dx4
dx2
(6)
and find its general solution.
2. Show that the solution pinned at x = 0 and satisfying w(z) = 0 has the form
w(x) = A (z sin(kx) − x sin(kz)) .
Show that the solution fixed at x = 0 and satisfying w(L) = 0 has the form
w(x) = A (kL − sin(kL))(cos(kx) − 1) − (cos(kL) − 1)(sin(kx) − kx) .
(7)
(8)
3. Use the previous solutions to find the critical values of k at which a nontrivial solution
(the “first buckling mode”) appears in
(a) beam with both ends pinned,
(b) beam with both ends fixed,
(c) beam with one end pinned and one end fixed.
Calculate the critical pressure assuming E = 1, I = 1 and L = 1. Plot the solution w(x)
in every case. Compare to the model at http://commons.wikimedia.org/wiki/File:
Buckledmodel.JPG
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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. Draw the phase portrait of the system by plotting some trajectories in Matlab.
4. An external factor, such as culling foxes, can shift the system from one trajectory to
another. In order to ensure the species’ survival you do not want to shift to a trajectory
with lower minima of F (t) and R(t). For which time interval of the oscillatory motion
should you issue fox hunting permits?
5. (*) Prove that the motion is periodic (hint: find a conserved quantity).
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