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Math Help - traffic flow

  1. #1
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    traffic flow

    Hello all:

    I have been studying a little traffic flow because of having been in road construction I never dreamed it was so involved, mathematically.

    Here's a problem:

    Suppose we are interested in the change in the number of cars N(t) between two observers, one fixed at x=a and the other moving in some prescribed manner, x=b(t).

    N(t)=\int_{a}^{b(t)}{\rho}(x,t)dx

    The derivative of an integral with a variable limit is

    \frac{dN}{dt}=\frac{db}{dt}{\rho}(b,t)+\int_{a}^{b  (t)}\frac{{\partial}{\rho}}{{\partial}t}dx

    Show this result using either considering:

    \lim_{h\to 0}\frac{N(t+h)-N(t)}{h} or by using the chain rule for derivatives.

    Also, Using \frac{{\partial}{\rho}}{{\partial}t}=\frac{{-\partial}}{{\partial}x}({\rho}u), show that

    \frac{dN}{dt}={-\rho}(b,t)\left[u(b,t)-\frac{db}{dt}\right]+{\rho}(a,t)u(a,t)

    I must admit, I am rather stymied. I ahve been going thorugh a rough time and I am having a difficult time concentrating.
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  2. #2
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    The first part is just Leibniz's Rule right? For the second, if:

    \frac{dN}{dt}=\frac{db}{dt}{\rho}(b,t)+\int_{a}^{b  (t)}\frac{{\partial}{\rho}}{{\partial}t}dx

    and:

    \frac{{\partial}{\rho}}{{\partial}t}=\frac{{-\partial}}{{\partial}x}({\rho}u)

    then:

    \frac{dN}{dt}=\frac{db}{dt}{\rho}(b,t)-\int_{a}^{b(t)}\frac{\partial}{\partial x}\left(\rho u\right)dx

    \frac{dN}{dt}=\frac{db}{dt}\rho(b,t)-\rho u\Biggr|_a^{b(t)}

    =\frac{db}{dt}\rho(b,t)-\rho(b,t)u(b,t)+\rho(a,t)u(a,t)
    Last edited by shawsend; December 3rd 2008 at 04:05 PM.
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  3. #3
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    Thank you. It is rather obvious now.I haven't been myself lately.
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