1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.


    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Determ the stationary temp with a PDE
  1. Stationary Values (Replies: 2)

  2. Stationary Points (Replies: 2)

  3. Stationary points (Replies: 9)