Define so .
The characteristic line is (See PDEMOC14).
But in the middle region because the characteristics determine that is constant with respect to .
Now and imply
Now we have the solution
The shock is at .
Here refers to the right limit for which is 0 and is the left limit which is 1. The shock line will determine where these regions meet and the constant solutions on either side of this line is the weak solution.
Find the shock condition
Plug in the shock point found earlier and let .
So now .
The weak solution is .