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Non-Homogeneous Heat Equation (Insulated Bar Question)

  1. Jun 7, 2010 #1
    1. The problem statement, all variables and given/known data
    Find U(x,t)

    dU/dt = d2U/dx2 + sin x

    Boundary Conditions:
    dU/dx (0,t) = 0


    U(1,t) = 0

    Initial Condition: U(x,0) = cos 7*π*x

    2. The attempt at a solution

    I start off with:
    d2(Un)/dx2 = λnUn (as an initial value problem)

    [d(Un)/dx](0) = 0; [d(Un)/dx](1) = 0

    My teacher told me to use the orthogonality later on, but at this point I'm stuck.
    Can anyone enlighten me? Thanks!
  2. jcsd
  3. Jun 7, 2010 #2


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    Homework Helper

    ok, so so far you have assumed separatino of variables, that is u(x,t) = U(x)T(t) and

    so now you'll need to find the eigenfunctions of the homogenous boundary value problem
    [tex] U_n''(x) = \lambda_n U_n(x) [/tex]
    [tex] U_n(0) = 0 [/tex]
    [tex] U_n'(1) = 0 [/tex]

    now solve the ODE above for 3 separate cases λn <0, =0, >0 and see which values of λn are allowable for the given boundary conditions. This yields the eigenvalues & eigenfunctions for the boundary value problem.
  4. Jun 7, 2010 #3
    I get like [tex] U_n = cos(n \pi x) [/tex] and [tex] \lambda _n = -(n \pi)^2 [/tex]
    Is that correct?
  5. Jun 7, 2010 #4


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    Homework Helper

    do they satisfy you boundary conditions?

    look ok for the 2nd, not so sure about the first
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