usual conservation equation Pt + 9x = 0, where su instead of taking the velocity of cars to be a linear fun functional form: u(p) = (1 - p)². (a) At what value of density p does the flux reach (b) What is the wave speed as a function of densit

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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7
Consider the dimensionless model for traffic flow with density p(x, t) and flux q(p) satisfying the
usual conservation equation Pr + qa = 0, where subscripts refer to partial derivatives. However,
instead of taking the velocity of cars to be a linear function of their density, let us assume a quadratic
functional form: u(p) = (1 – p)².
(a) At what value of density p does the flux reach a maximum?
(b) What is the wave speed as a function of density?
1 to the left of the origin, and p
(c) For the initial condition where p =
and sketch the characteristics in the expansion fan region (emanating from the origin) in the
(x, t)-plane.
= 0 to the right, find
Transcribed Image Text:Consider the dimensionless model for traffic flow with density p(x, t) and flux q(p) satisfying the usual conservation equation Pr + qa = 0, where subscripts refer to partial derivatives. However, instead of taking the velocity of cars to be a linear function of their density, let us assume a quadratic functional form: u(p) = (1 – p)². (a) At what value of density p does the flux reach a maximum? (b) What is the wave speed as a function of density? 1 to the left of the origin, and p (c) For the initial condition where p = and sketch the characteristics in the expansion fan region (emanating from the origin) in the (x, t)-plane. = 0 to the right, find
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