Suppose that a solution to Burgers Equation exists as a density wave moving with

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Suppose that a solution to Burgers Equation exists as a density wave moving without change of shape: rho(x, t) = f(x - Vt) at velocity V. What Ordinary Differential Equation is satisfied by f? Integrate this ODE once. By graphical techniques, show that a solution exists such that f rightarrow rho2 as x rightarrow + infinity and f rightarrow rho1 as x rightarrow -infinity, only of rho2 > rho2. Give a rough sketch of this solution and an interpretation of the the result in terms of traffic. Show that the velocity of wave propagation, V, is the same as the shock velocity separating rho = rho 1 from rho = rho 2, occurring if nu = 0. For this problem we will begin with the conservation of cars equation where rho := traffic density, u(rho) := ca velocity, and q(rho) := rho u(rho) = car flux, such that c(rho) = q'(rho). Only now we will enrich the model by allowing, instead of the simple u = u(rho) model above, the new model u = U(rho) - nu/rho rho/ x where nu is some constant. What sign should nu have for the above expression to be reasonable based on what were trying to model (traffic)? What is the new conservation of cars equation? Assume U(rho) = umax (1 - rho/rho max). Derive Burgers Equation:

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