MTH 323 – Mathematical Modeling Sample Final Exam
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MTH 323 – Mathematical Modeling
Sample Final Exam
Answer each of the following:
1. What is a mathematical model?
2. What is a conceptual model?
3. What is a mechanistic model?
4. What is a quantitative model and when would it be preferable over a conceptual model?
5. Consider a mass-spring system assuming Hooke’s Law.
(a) How does the period change in the solution to an undamped system if mass is quadrupled?
(b) What is the qualitative difference between solutions to an overdamped system versus an underdamped system?
(c) How might the solution to an overdamped system change if the mass is increased?
(d) Is resonance possible with discontinuous forcing?
6. Consider the pendulum
model
L = -g sin(θ).
(a) Linearize the pendulum model.
(b) State assumptions under which the linearization is valid.
(c) Solve the linearized model using initial conditions θ(0) = and (0) = 0.
(d) Does your solution fit the validity assumptions?
7. Consider the discrete logistic model
Nm+1 = ρNm ╱ 1 - 、 .
(a) Non-dimensionalize by setting U = N/k .
(b) What are the equilibria of the non-dimensionalized model?
(c) Give the stability criteria for one of these equilibria.
(d) What qualitative behavior may be exhibited around carrying capacity in a discrete logistic model that is not possible in the ODE logistic model?
8. Consider an age-structured population with three classes: N0 (t), N1 (t), N2 (t).
(a) Assuming birth rates given by b1 , b2 (youngest class does not reproduce), and survival rates s0 , s1 (oldest class does not survive). Write down the Leslie matrix model for the population dynamics.
(b) How is the population at the kth time step related to the initial population?
(c) Assuming that a dominant eigenvalue of the matrix exists, what can you say about the population distribution in the limit as k → o?
(d) What values of a dominant eigenvalue would lead to the total population dying out?
Consider the traffic flow model
+ (ρu) = 0 x e R, t > 0
where u is velocity, ρ is density, umax and ρmax are known constants, and we assume
u(ρ) = umax ╱ 1 - 、
9. (a) Describe physically what the linear velocity-density relation implies (may be helpful to draw the graph).
(b) Recall that q = ρu is the flow rate. What is the maximum flow rate for this model (i.e., the capacity of the road) and at what density does it occur?
(c) In a heavy traffic density scenario what can you say about the density wave velocity?
10. Assume ρmax = 1 and umax = 1 (non-dimensionalized model). Sketch the characteris- tics for the solution to the PDE model given the following initial condition [hint: first determine q\ (ρ) away from discontinuities]
ρ(x, 0) = ,0(1)
if |x| < 1,
otherwise .
2022-12-07