School of Mathematics and Statistics
MAST30030
Applied Mathematical Modelling
Assignment 2. Due: 5pm on Friday 29 April 2022
This assignment counts for 5% of the marks for this subject.
Consider an infinitely long highway, containing a uniform distribution of cars that are moving in the
positive x-direction. Careful measurements show that the car velocity v obeys the relation v = 1−ρ,
where ρ is the linear density of cars along the highway. Both v ∈ [0, 1] and ρ ∈ [0, 1] are considered
to be the non-dimensionalised variables. A slip lane exists in the region 0 ≤ x < 1 where cars join
the highway at a constant rate of β cars per unit length per unit time. This slip lane is closed for all
time t < 0, and then opens for t ≥ 0.
(a). Write down the conservation equation involving the density of cars, ρ(x, t), the flux of cars,
J(x, t), and the rate of cars entering the highway.
(b). Suppose that the initial traffic density ρ(x, 0) = ρ0 > 1/2, so that the flux is below the max-
imum value (heavy traffic). Calculate the characteristics emanating from t = 0 within the
region 0 < x < 1, and find the maximum value of β = βC such that the traffic density remains
less than one for all times. Henceforth assume that β < βC .
(c). Calculate the characteristics emanating from outside the slip lane (i.e., x < 0 and x > 0 at t =
0). Calculate the value(s) of ρ(x, t) at both ends of the slip lane, i.e., find ρ(0, t) and ρ(1, t).
(d). Identify the presence of key features of this problem, including shocks/fans. For any shocks
beginning from t = 0, find a differential equation that governs the position of the shock front,
but do not solve.
(e). Show that after some critical time, t∗ (you do not need to find this value explicitly), there is
a shock behind the slip lane (x < x∗ < 0), propagating with constant negative velocity V =
dxs/dt. Find the value of V .
(f). Draw the full space-time diagram showing all characteristics, fan(s), shocks(s) (where applica-
ble).
(g). Sketch the density profile ρ(x, t) at two different time points, 0 < t < t∗, and t > t∗. Describe
the long term behaviour of the system, and reconcile this with the fact that ρ0 > 1/2.
(h). What is the flux of cars in the region x > 1? Explain why this does/does not depend on the
value of β.
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