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Laplace equation w/ dirichlet boundary conditions - Partial Diff Eq.

  1. Nov 8, 2012 #1
    1. The problem statement, all variables and given/known data

    The steady state temperature distribution [tex]T(x,y)[/tex] in a flat metal sheet obeys the partial differential equation:

    [tex]\displaystyle \frac{\partial^2 T}{\partial x^2}+ \frac{\partial^2 T}{\partial y^2} = 0 [/tex]

    Seperate the variables in this equation just like in the one-dimensional wave equation and find T everywhere on a square flat plate of sides S with the boundary conditions:

    [tex] T(0,y)=T(S,y)=T(x,0) , T(x,S)=T_0 [/tex]

    2. Relevant equations



    3. The attempt at a solution

    this is what I have so far..not sure what to do next???

    [url=http://postimage.org/image/k0ujzisqz/][PLAIN]http://s17.postimage.org/k0ujzisqz/steady_state_heat_prob.jpg[/url][/PLAIN]
     
    Last edited: Nov 8, 2012
  2. jcsd
  3. Nov 8, 2012 #2
    I'm not bumping my post, sorry! I forgot to mention that classmates mentioned [tex]sinh[/tex] shows up in the fourier series and I don't know why also. Any help getting started would be greatly appreciated

    many thanks
     
  4. Nov 9, 2012 #3

    vela

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    The differential equation for Y(y) has a sign mistake in it, though the solutions you wrote down are correct. Instead of exponentials, you can write down the solution in terms of the hyperbolic functions
    \begin{align*}
    \cosh ky &= \frac{e^{ky}+e^{-ky}}{2} \\
    \sinh ky &= \frac{e^{ky}-e^{-ky}}{2}
    \end{align*} It's analogous to how you can write down the solutions for X(x) in terms of sin kx and cos kx or in terms of eikx and e-ikx. You're just choosing a different basis.

    Now you have to apply the boundary conditions.
     
  5. Nov 9, 2012 #4
    Thanks vela !
     
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