Consider the first-order nonlinear partial differential equation
ut+uux=0
on t >0, - < x < o,with initial value given by
2x<-1
u(x,0)=
x>-1.
1. 1 What is the speed of the wave? Justify
2. (2) A shock will appear. Why would you expect this?
3. 5 Determine the characteristic curves in terms of the initial condition uo,t at x0=xo
4. (5) Draw these characteristics curves in (x,t)-axes, i.e., in the horizontal axis and t in the vertical axis.
2
5.(5) Show that the speed of the shock, dxs/dt, is the average of the two initial speeds.
6. (5) Determine when and where the shock first appears.
7.(5) What is the path of the shock xs=s(t)? Justify
8.(5) Assuming s(t = ct + c2,with ci and c real constants,write the solution u = u(, t) of the PDE (if you solved for the path of the shock, s = s(t), you can use your solution).