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Heat equation

  1. Feb 10, 2009 #1
    1. The problem statement, all variables and given/known data
    http://img3.imageshack.us/img3/5020/84513876dm0.png [Broken]


    3. The attempt at a solution
    I found that [tex] f(t) =exp \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right) [/tex]

    Is this correct?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Feb 11, 2009 #2
    I thus only solved the ODE -problem.
     
  4. Feb 11, 2009 #3

    cristo

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    Show your working, and it will be easier to check whether you are correct or not.
     
  5. Feb 11, 2009 #4
    OK, you got it:

    [tex]\frac{ \partial{u} }{ \partial{t} } = \frac{df}{dt} \sin(\frac{m \pi x}{L}) [/tex]

    [tex]\frac{ \partial{^2 u} }{ \partial{x^2} } = f(t) \left( \frac{-m^2 \pi ^2 }{L^2} \right) \sin(\frac{m \pi x}{L}) [/tex]



    Therefore

    [tex]
    f(t) =\mbox{exp} \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right)
    [/tex]


    Conclusion:
    [tex]
    u(x,t)= u_0 + \mbox{exp} \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right) \cdot \sin(\frac{m \pi x}{L})[/tex]
     
  6. Feb 11, 2009 #5

    Redbelly98

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    Looks good, I'm just wondering one thing.

    Should ρ, c, and k be in there somewhere, or do we assume units such that those are all =1?
     
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