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Heat Equation

  1. Feb 2, 2009 #1
    1. The problem statement, all variables and given/known data

    If there is heat radiation in a rod of length L, then the 1-D heat equation might take the form:

    u_t = ku_xx + F(x,t)

    exercise deals with the steady state condition => temperature u and F are independent of time t and that u_t = 0.

    u_t = partial derivative with respect to t
    u_xx = 2nd partial derivative with respect to x

    Problem: find u(x) if F(x) = sinx, k = 2, u(0) = u'(0), u(L) = 1

    2. Relevant equations


    e^(itheta) = cos(theta) + isin(theta)
    e^(-itheta) = cos(theta) - isin(theta)
    e^(itheta) + e^(-itheta) = 2cos(theta)
    e^(itheta) - e^(-itheta) = 2isin(theta)

    3. The attempt at a solution

    u_t = ku_xx + F(x,t)
    u_t = ku_xx + F(x)G(t)
    2u_xx + sin(x)G(t) = 0
    characteristic equation: m^2+1=0, m^2 = -1, m = +-i
    general soltn: u(x) = c1cos(x) + c2sin(x)
    u(0) = c1cos(0) + c2sin(0)
    u'(x) = -c1sin(x) + c2cos(x)
    u'(0) = -c1sin(0) + c2cos(0)
    => c2 = c1
    u(x) = c1cos(x) + c1sin(x)
    u(L) = c1cos(L) + c1sin(L) = 1

    from here I'm actually a little lost, I'm not sure what to do with u(L) = 1 portion; do I perhaps need to find a particular solution? any help would be appreciated.
     
  2. jcsd
  3. Feb 2, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If everything is time independent then it's really just an ODE with boundary conditions, right? Yes, find a particular solution. Then find a homogeneous solution and adjust the constants to fit the boundary conditions.
     
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