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Homework Help: Determ the stationary temp with a PDE

  1. May 18, 2006 #1
    I have a square area with the length a. The temperature surrounding the square is T_0 except at the top where it's T_0(1+sin(pi*x/a)). They ask for the stationary temperature in the area. In other words, how can the temperature u(x,y) inside the area be written when the time = infinity.

    The first thing I do is that I realize that u(x,y) can be written u(x,y) = sum(X(x)*Y(y)).
    I also think that it should be a nice starting point to create v = u-T_0, that gives me that the surrounding temperature is 0 everywhere except at the top where it's T_0*sin(pi*x/a)

    experience tell me that X(x) = sin(k*pi*x/a)

    But how do I find Y(y)? cant seem to get it right.
     
  2. jcsd
  3. May 26, 2006 #2
    Start with the heat equation in 2dimensions, with du/dt=0 because you want the stationary solution. Let u(x,y)=X(x)Y(y) and put it into the p.d.e.
    Then you separate variables.

    [tex]X''/X=-Y''/Y=-\lambda^2[/tex]

    with cos/sin and exponential solutions. The homogenous boundary conditions eliminatate some of the constants.

    and you end up with something like

    [tex]u(x,y)=\sum_{m=0}^\infty B_m*sinh(m\pi y)*sin(m\pi x)[/tex]

    [tex]u(x,a)=\sum_{m=0}^\infty[B_m*sinh(m\pi a)]*sin(m\pi x)[/tex]

    where B_m*sinh(mpia) are the coefficients in a fourier sin-series
     
    Last edited: May 26, 2006
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