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Heat Equation with insulated endpoints.

  1. Oct 25, 2009 #1
    1. The problem statement, all variables and given/known data
    Assume that a bar is insulated at the endpoints. If it loses heat through its lateral surface at a rate per unit length proportional to the difference u(x,t) - T, where T is the temperature of the medium surrounding the bar, the equation of heat propagation is now

    [tex]u_{t} = k u_{xx} - h (u-T)[/tex]

    where h > 0
    2. Relevant equations
    Use the function

    [tex] v = e^{ht}(u-t) [/tex]

    to reduce this BVP to one already solved.


    3. The attempt at a solution

    "To one already solved" is referring to heat equation variants in which the PDE is of form

    [tex]u_{t} = k u_{xx} [/tex]

    I can solve it from that form, I just need to convert into something of that form.

    Some things I noticed, partial derivative of v with respect to t, and equated to the second partial derivative with respect to x yields u_t = u_xx - h(u-t)
    This is off by the constant k which is in front of the u_xx in the original PDE. Not sure what I'm missing from here.
     
  2. jcsd
  3. Oct 26, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi Kizaru! :smile:

    (try using the X2 and X2 tags just above the Reply box :wink:)
    Yes, you have the correct basic technique, I can't see quite how you haven't got there. :confused:

    If v = eht(u - t),

    then vt = … and kvxx = … ? :smile:
     
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