EGN 5456 Homework 5 Fall 1996

1. Solve
   
   
   
   
2. Draw the solution of the previous question in the u,x-plane for times t=1, 2, and .
3. Draw the characteristic lines of the partial differential equation of the previous questions. Indicate along which of these characteristics the solution is singular.
4. If we replace the initial condition in the previous questions by u(x,1)=f(x) where f is smooth, (eg ), will there ever be singularities in u? Explain.
5. One-way vehicular traffic satisfies the continuity equation of fluid dynamics,
   
   Is this in conservation law form? Is it in divergence form?
6. The previous equation can be differentiated out to give
   
   Is this in conservation law form? Is it in divergence form?
7. Find a conservation law for any arbitrary finite road interval by integrating the differential conservation law between a and b.
8. Explain the conservation law you got in the previous question physically.
9. Plot the characteristics of the traffic equation assuming that the velocity and that the initial density at time t=0 is given by .
10. Solve the problem of the previous question for in terms of and t, where is the starting x-position of the characteristic at time t=0/
11. Plot the car density in the -plane for times t=0, t=.25 and t=.5.
12. Find the locations and times at which the first car collisions (shocks) occur.
13. The Euler equations of 2D steady, inviscid, nonconducting two-dimensional flow are:
    
    
    
    
    
    
    
    Show that according to the last equation
    
    where .
14. Classify the system of the previous question using the fifth equation instead of the fourth (entropy) one.
15. Classify the following system of equations:
    
    
    
    
16. Find and sketch the characteristic lines of the equations of the previous question.
17. Find the compatibility equations equivalent to the system of equations.
18. Find the Riemannian invariants of the compatibility equations assuming the initial conditions , .
19. From the Riemannian invariants, find the solution u(x,t) and v(x,t) to the system of P.D.E.s and initial conditions.

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