AM107 | AM107
Takehome Exam (Problem 3)
Handed out: Thursday, November 30, 2000 Return by: 5:00pm on Thursday, December 14, 2000 |
Problem 3: Consider the model
a. Moving to a traveling frame, let U(z)
= u(x,t) and V(z) = v(x,t) where z = x - ct. Write down the two higher-order
ODE's corresponding to the PDE's above written in this new frame.
b. Usually, we would introduce a new
variable W = Vz and write the equations in a. as a system of
three first order equations in U, V and W. Instead, add the
equation for U to the equation for V and note the cancellation of the UV-terms.
c. Integrate this equation once to
get a single first order equation for Vz. (Don't forget
the constant of integration....)
d. Now we have a two-dimensional system.
Find the steady states and evaluate their stability (as a function of a
and c) to determine whether or not there may exist a trajectory connecting
the two steady states.
e. Use phase-plane techniques (or pplane5)
to verify the existence of this connecting orbit for appropriate values
of the parameters. Plot U and V vs. z for this trajectory to exhibit
the qualitative shape of this traveling wave (pplane5 can do this too).
f. Hypothesize about what sort of system
these equations might model.