 # Applied Differential Equations

## Transcribed Text

1. The standard traffic model for the car density ρ(x, t) is ρt + Fx = 0 where v = 1 − ρ F(ρ) = ρv = ρ − ρ 2 Write down the strong solution to the initial-value problem with ρ(x, 0) = A(x) and find the exact solution for the straight line initial conditions ρ(x, 0) = αx with constant α. 2. For the standard traffic model from question 1, the characteristic wave speed c(ρ) = F 0 (ρ) = 1 − 2ρ. Show that for strong solutions the Burgers equation holds for c(x, t), i.e., ct + ccx = 0. Are the week solutions of this equation identical to those from question 1? (Hint: you need to compare the shock speeds to answer this question.) 3. Draw the graph of F(ρ) for equation from question 1 and give a graphical interpretation of the shock speed formula cs = F(ρR) − F(ρL) ρR − ρL in terms of the slope of a secant. Also give a graphical interpretation of the Lax stability criterion c(ρL) ≥ cs ≥ c(ρR) using the same graph. This allows determining the stability of shocks as a function of the pairs (ρL, ρR). 4. A useful highway driving safety guideline is to keep a distance to the car in front of you that is proportional to your speed, so 1/ρ ∝ v. A simple model of this rule (including a speed limit) is v(ρ) = ( 1 0 ≤ ρ ≤ 1 1/ρ 1 < ρ Here ρ ∈ [0,∞] is allowed. Compute and sketch F(ρ) and c(ρ) for this model. For strong solutions, what does this model predict for the initial-value problem in which ρ(x, 0) is specified? (Hint: you need to distinguish between characteristics on which ρ is bigger or smaller than unity.) Compute the shock speed for this model and use the Lax criterion to find the stability of shocks. (Hint: it is easiest to use the graphical construction from the previous questions.)

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