Ross Ogilvie      23rd September, 2024
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10.

Don’t cross the streams Consider the PDE ${\partial }_{x}u+x{\partial }_{y}u=0$ on the domain $y>0$ with the boundary condition $u(x,0)=g(x)$.

(a)

Show that the boundary hyperplane $\{y=0\}$ is non-characteristic at $(x,0)$, except for $x=0$. (1 point)

(b)

What condition does the PDE impose on the boundary data $g$ at the point $(0,0)$? (1 point)

(c)

Determine the characteristic curves of this PDE. (1 point)

(d)

By considering the $y$-derivative of $u$ on the boundary hyperplane, show that there is no $C^1$ solution with the initial data $g(x)=x$. (2 points)

11.

It’s just a jump to the left

In this question we explore some other solutions to the initial value problem from Example 1.10. As we saw, for small $t$ the method of characteristics gives a unique solution

$$u_{t<1}(x,t)=\{\begin{array}{cc}1\hfill & \text{for }x<t\hfill \\ \frac{x-1}{t-1}\hfill & \text{for }t\le x<1\hfill \\ 0\hfill & \text{for }1\le x.\hfill \end{array}$$

(a)

(Optional) Derive this solution for yourself, for extra practice.

After $t=1$, the characteristics begin to cross and so the method cannot assign which value $u$ should have at a point $(x,t)$. However, we could still arbitrarily decide to choose a value of one characteristic. Consider therefore

$$v(x,t)=\{\begin{array}{cc}{u}_{t<1}\hfill & \text{for }t<1\hfill \\ 1\hfill & \text{for }x<t\hfill \\ 0\hfill & \text{for }t\le x\hfill \end{array}$$

(b)

Draw the corresponding characteristics diagram in the $(x,t)$-plane for this function.
(2 points)

(c)

Describe the graph of discontinuities $y(t)$. Compute the Rankine-Hugonoit condition for $v$. (2 points)

(d)

How much mass (i.e. the integral of $v$ over $x$) is being lost in the system described by $v$ for $t>1$? (2 points)

12.

You’re not in traffic, you are traffic

In this question we look at an equation similar to Burgers’ equation that describes traffic. Let $u$ measure the number of cars in a given distance of road, the car density. We have seen that $f$ should be interpreted as the flux function, the number of things passing a particular point. When there are no other cars around, cars travel at the speed limit $s_m$. When they are bumper-to-bumper they can’t move, call this density $u_m$.

(a)

What properties do you think that $f$ should have? Does $f(u)=s_{m}u\cdot (1-u∕u_m)$ have these properties? (2 points)

(b)

Find a function $f$ that meets your conditions, or use the $f$ from the previous part, and write down a PDE to describe the traffic flow. (1 point)

(c)

Find all solutions that are constant in time. (2 points)

(d)

Consider the situation of the start of a race: to the left of the starting line, the racecars are queued up at half of the maximum density (ie $0.5u_m$). To the right of the starting line, the road is empty. Now, at time $t=0$, the race begins. Give a discontinuous solution that obeys the Rankine-Hugonoit condition, as well as a continuous solution. (4 points)