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I A few questions about Green's Functions...

  1. Feb 12, 2017 #1
    Hi, I am having some trouble understanding exactly when a modified green's function is needed. Here is the general problem:

    [tex] Lu = (p(x)u'(x))' + q(x)u(x) = f(x), x_0 \leq x \leq x_1, p(x) > 0,\\
    \alpha_0 u(0) + \beta_0 u'(0) = 0, \alpha_1 u(1) + \beta_1 u'(1) = 0[/tex]

    In my notes it says if the corresponding homogeneous problem with the same boundary conditions has a non-zero solution [itex] u^*(x) [/itex], then we can construct a standard Green's function, with the solvability condition [itex] \int_{x_0}^{x_1} u^*(x)f(x)dx = 0 [/itex]. Then later in my notes it says if [itex] u^*(x) [/itex] is a non-zero solution, we need to use a modified Green's function. Which one is it?

    Another question I have is regarding the formula involving the Wronskian, if we have to linearly independent solutions,

    [tex] G(x,z) = -\dfrac{u_1(x)u_2(z)}{p(x)W(x)}, x_0 \leq z \leq x,\\
    -\dfrac{u_1(z)u_2(x)}{p(x)W(x)}, x \leq z \leq x_1 [/tex]

    When exactly is this solution valid?

    Thank you.
     
  2. jcsd
  3. Feb 19, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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