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Sturm-Louville problem: u'' - u = x, u(0)=u(π)=0.

  1. Apr 30, 2012 #1
    1. The problem statement, all variables and given/known data

    screen-capture-5-10.png

    2. Relevant equations

    From the lecture notes:

    "To study the Sturm-Louiville operator Ls in greater detail, we first need to determine it's adjoint operator, denoted by Ls. [...] The adjoint of the Sturm-Louiville operator satisfies the relationship

    <Lsy1,y2>=<y1,Lsy2>,​

    where y1 and y2 are two arbitrary functions satisfying the prescribed boundary conditions."

    3. The attempt at a solution

    I went straight ahead and tried to find the adjoint of the Sturm-Louiville operator. It got really messy, and I'm wondering whether there's something obvious that I'm missing.

    screen-capture-6-6.png
     
  2. jcsd
  3. May 3, 2012 #2
    First, the operator Lu=u''-u≠x in general, otherwise there's no need to solve the DE.
    Second, the boundary condition u(0)=u(pi)=0 applies to all u, v, w, etc. By letting <Lu,v>=<u,Ad{L}v>, and using integration by parts, it can be shown Ad{L}=L.
     
  4. May 4, 2012 #3
    Got it.

    Now the next part says

    Well, since L*v = Lv = v '' - v, we need to solve v'' - v = 0 with v(0) = v(pi) = 0. Isn't the only solution v(x) = 0, since v '' - v = 0 implies v = C1ex + C2e-x????/
     
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