Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Math 412 HW 1

You must show your work and explain your reasoning to obtain full credit. Turn the assignments in on Canvas.

1. Here we’ll derive the PDE for traffic flow that we’ll work with later in the course. Solving it will be completely different from the heat equation, but the derivation is quite similar to what we did in class for the heat equation so it will be interesting for you to work through these steps now.

Here we consider one lane of traffic on a straight highway occupying 0 ≤ x ≤ L.

Let ρ(x, t) be the traffic density, in cars per km, at position x and time t. This is our dependent variable / unknown function we are interested in modeling.

Let φ(x, t) be the rightward traffic flux, in cars per hour, at position x and time t.

We won’t consider sources/sinks here for simplicity, but you could consider that to represent on/off ramps, intersections, etc.

(a) Consider an arbitrary segment of the highway, a ≤ x ≤ b. Write an expression for the total number of cars in this segment.

(b) Write down the equation expressing “conservation of cars”. Recall in class, we used the fact that the conservation of heat energy meant that the change in total heat in a segment of the rod was the net contribution of flux through the ends and source terms inside.

(c) Recall in class, we needed to use physics (Fourier’s law in this case) to express the flux φ in terms of the dependent variable in order to obtain a “closed system of equations”. Convince yourself that

φ(x, t) = ρ(x, t)v(x, t),

where v(x, t) is the (rightward) traffic speed, in km per hr, at position x and time t. We will now do our “traffic science”: assume there is a “constitutive relation” v = v(ρ) that captures drivers’ tendency to drive faster when there is less traffic. The simplest assumption would be to choose v(ρ) linear such that v(0) = vmax > 0 is the top speed drivers will have in the case of zero traffic density; and the existence of a “bumper-to-bumper traffic jam density” satisfying v(ρmax) = 0.

Use this to write φ in terms of ρ (and the given parameters above), and insert this into your “conservation of cars” equation.

(d) Follow the steps from class to obtain the integral conservation law, and obtain the partial differ-ential equation for ρ(x, t) by noting the segment under consideration a ≤ x ≤ b was arbitrary.

2. Do problems 1.2.3, 1.2.5, 1.2.8. (These are the same problems in both the 4th and 5th editions — I’ll be sure to check each week if there are any differences.)

3. Relevant Math 250 review for next week: Consider the second order constant coefficient ODE

Write the general solution for each of the three cases:

(a) λ > 0,

(b) λ = 0,

(c) λ < 0.