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1.1 Suppose that p(I.t) is the number density of cars evolving according to the traffic model 8p a(pu) 0. at ax with u the car speed. Take u=1-p. A queue is building up at traffic light so that, when the light turns green at t = 0, p(x.0) Show that initially the characteristics, which are straight lines, are in xt+l, where p=0, 2s)t where -2s)f in 1-t Deduce that collision first occurs at 1/2 when 1/2, and that thereafter there is a shock that 2t 1.2 Consider the problem of thin layer of paint of thickness h(x.t) and speed U/I .t) flowing down a wall, see the following figure. The paint is assumed to be uniform in the z-direction. The balance between gravity and viscosity (fluid fiction) means that the y surface of paint layer h(x,t) wall u(x,y,t) Y x velocity satisfies the equation ara dy where cisa positive constant. This is supplemented by the boundary conditions au(x.y.t) u(x,0,1) 0. =0. dy The density of paint per unit length in the r-direction is ph(z.t) where Po is constant, and the corresponding flux is a) Using conservation of paint, and solving for u(z.y.t) in terms of h(I,t) and y, derive the following PDE for the thickness h: ah ah at az b) Set = 1. Show that the characteristics are straight lines and that the Rankine- Hugoniot condition on shock S(t) is dS [3/3]+ dt A stripe of paint is applied at 0 so that h(x,0) Show that. for small enough t, (z/t)¹², Oh = where the stock is S(t) /3. Explain why this solution changes at 3/2,and show that thereafter dS s di 3t 1.3 Look for traveling wave solution U/I. t) U(z) with (X Vt)/e, to the equation ax for DO. Show that du U= VU constant. dz n 2 and deduce that V [Un/11/20 Uto Discuss how the traveling wave solution relate tostock solutions of the quasilinear equation obtained by setting o. Also show that. when n = 2.U only tend U(00) can to as z 00 if dU/dz 1.4 Traffic in tunnel A reasonable model for the car speed in a very long tunnel is v(p) where 1- In(pm/pc) Assume the entrance is at 0 and the cars are waiting (with maximum density) the tunnel to open to traffic at time 0. Thus. the initial condition is a) Determine the density and car speed and illustrate how their profiles as a function of change with time. b) Determine and draw in the (x t) plane the trajectory of car initially at <0 and compute the time it takes to enter the tunnel. 1.5 Solve Burger's equation uz wu. =0 with initial data g(x) (0,I>1 1.6 Using the Hopf Cole transformation, solve the following problem for the viscous Burger equation where H(x) is the Heaviside function

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Differential Equations Differential Equations
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