Consider the case where $a(u)=(1-u)$ so that $f(u)=u(1-u).$ This is known as

the Greenshields flux model in traffic flow. It arises in the development of fundamental diagrams

relating traffic density to the flow rate. The initial condition is $u(x,0)=u_0(x).$ This is Burgers'

equation.

a$.$ What is $\frac{dx}{dt}?$

b$.$ If the initial condition is defined as in $(3$a$)$ above, roughly sketch the characteristic lines.

Labe$\frac{l}{c}$allout the shock and rarefaction parts of the sketch.

c$.$ What is the initial flux $($flow rate$)$ for the segment $0x1?$ What does this physically

represent?

d$.$ From which side does the shock propagate, and is this reasonable? Please explain