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
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
Section: Chapter Questions
Problem 1RQ
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Question
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff38c564c-0ec3-4396-947c-6e6b11243ead%2Ff3dc6008-4694-4915-b403-f939b2ae2efd%2Farvjz2g_processed.jpeg&w=3840&q=75)
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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