Introduction to Partial Differential Equations
Exercise Sheet 4

Ross Ogilvie      23rd September, 2024
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10.
Don’t cross the streams Consider the PDE xu + xyu = 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 C1 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

ut<1(x,t) = { 1 for x < t x1 t1 for t x < 1 0 for 1 x.
(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) = { ut<1for t < 1 1 for x < t 0 for t x
(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 sm. When they are bumper-to-bumper they can’t move, call this density um.

(a)
What properties do you think that f should have? Does f(u) = smu (1 uum) 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 0.5um). 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)