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Desperate trying to solve a simple Sturm-Liouville equation

  1. Sep 30, 2011 #1
    While dealing with a wave problem,
    I encountered the following equation

    [tex] \frac{d}{dx}\left[(1-x^2)^2\frac{d}{dx}y\right] - k^2y = -\omega^2y [/tex]

    with x ∈ [0,A], (0<A<=1)

    where k is a real number.

    Thus it has eigenvalue ω^2 and weight unity.

    Boundary conditions are

    [tex] \frac{dy}{dx} = 0 [/tex]

    at x = 0 and

    [tex] y=2A[/tex]

    at x= A.


    I only need to obtain the solution for the ground state (the one with lowest eigenvalue).

    for general values of k>0, 0<A<1.

    I find from the physical point of view that the solution should look like

    y_0=constant for A->1,

    and y_0 = cosh(kx) for A -> 0


    Can anybody give me a hint on how to solve this equation?
     
    Last edited: Sep 30, 2011
  2. jcsd
  3. Oct 1, 2011 #2
    Here's a numeric approach in Mathematica:

    Code (Text):

    k = 1.21;
    \[Omega] = 3.25;
    a = 0.825;

    sols =
       (First[NDSolve[{(1 - x^2)^2*Derivative[2][y][
                x] - 4*x*(1 - x^2)*Derivative[1][y][
                x] + (\[Omega]^2 - k^2)*y[x] == 0,
            Derivative[1][y][0] == 0, y[a] == 2*a}, y,
           x, Method -> {"Shooting",
             "StartingInitialConditions" ->
              {y[0] == #1, Derivative[1][y][0] ==
                0}}]] & ) /@ {-2, 0, 2};

    Plot[Evaluate[y[x] /. sols], {x, 0, a},
      PlotStyle -> {Black, Blue, Green}]
     
     
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