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Applying Boundary Conditions

  1. Nov 21, 2008 #1
    1. The problem statement, all variables and given/known data

    Solve

    [tex] \left(\frac{\partial^{2}}{\partial z^{2}}- \frac{1}{\kappa} \frac{\partial}{\partial t}\right) T=0 [/tex]

    with boundary conditions

    (i) [tex] T(0,t)=T_{0}+T_{1} Sin[\omega t] [/tex]

    (ii) [tex] T [/tex] is finite as [tex]z\rightarrow\infty[/tex]


    2. Relevant equations

    Separation of Variables gives:

    [tex] T = Z(z) U(t) \Rightarrow \frac{Z''}{Z}=\frac{1}{\kappa} \frac{U'}{U}=- \lambda^{2}[/tex]

    This gives typical solution to the heat conduction equation:

    [tex]U(t)=A_{0} e^{-\lambda^{2} \kappa t}[/tex]

    [tex]Z(z)=B_{0} e^{i \lambda z} + B_{1} e^{-i \lambda z}[/tex]

    Now to satisfy (i) and (ii) consider

    [tex] \lambda^{2} = \frac{i \omega}{\kappa} [/tex]
    [tex] \lambda = \sqrt{\frac{ \omega}{2 \kappa}} (1+i) [/tex]

    so from (ii) [tex] B_{1}=0 [/tex]

    With some algebra aside, we have

    [tex] T(z,t)= C e^{-\sqrt{\frac{ \omega}{2 \kappa}} z} e^{-i \left(\omega t -\sqrt{\frac{ \omega}{2 \kappa}} z\right)} [/tex]


    3. The attempt at a solution

    Now originally, I tried

    Consider

    [tex] T(z,t)=e^{-\sqrt{\frac{ \omega}{2 \kappa}} z} ( C_{Re}Cos(\omega t -\sqrt{\frac{ \omega}{2 \kappa}} z)} - i C_{im} Sin(\omega t -\sqrt{\frac{ \omega}{2 \kappa}} z)})[/tex]

    [tex] T(0,t)= C_{Re} Cos(\omega t) - i C_{im} Sin(\omega t) = T_{0} + T_{1} Sin(\omega t)[/tex]

    so [tex] T_{Re}= T_{0} Sec( \omega t) [/tex] and [tex] T_{Im}= i T_{1} [/tex]

    But, I don't know I realized that (i) is a real number and that taking the real part of the solution instead and matching that up with (i), this would give me

    [tex] T(z,t)=C e^{-\sqrt{\frac{ \omega}{2 \kappa}} z} Cos(\omega t -\sqrt{\frac{ \omega}{2 \kappa}} z)} [/tex]

    with [tex] C= T_{0} Sec(\omega t) + T_{1} Tan(\omega t) [/tex]

    I am convinced one of them is wrong, which is my first attempt. I did some plots and stuff, and they behave too differently.

    Separation of Variables feels like more of an art. Took me awhile to come up with that eigenvalue because the solution made no sense at first like exploding time terms or non-damping spatial terms etc.
     
    Last edited: Nov 21, 2008
  2. jcsd
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