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Homework Help: Heat Equation Initial Conditions

  1. Feb 14, 2010 #1
    Greetings all,

    I have a question in regards to my initial conditions. The problem as given is:

    ut=uxx with u' = 0 at x=0 and u=0 at x=L

    I was also given u={1 0<x<L/2, 0 L/2<x<L

    I understand the set up of the problem and the solving of it for the most part, however I'm having trouble understanding the point that the third condition (the one about u) makes. More or less I'm stuck with trying to figure out what X is initially.

    Thank you
  2. jcsd
  3. Feb 14, 2010 #2


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    u' = 0 is ambiguous; this is a partial differential equation, and these are boundary conditions. I assume you mean:

    ux(0,t) = 0, u(L,t) = 0 for t > 0.

    Ditto here. Write it correctly:

    u(x,0) = 1, 0<x<L/2, 0 L/2<x<L

    Your two boundary values for x should give you an eigenvalue problem after you separate variables. The last condition, which is an initial condition, tells you the initial temperature of the bar at t = 0. You should be able to satisfy that using Fourier series from your eigenvalue problem.
  4. Feb 14, 2010 #3
    Thank you for your help. My teacher had written up our homework in the form that I wrote it in originally, and thus I was having trouble trying to figure out what the initial conditions were exactly. Your post clarified it, and I should be set to solve it now.

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