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Boundary Value Problem + Green's Function

  1. Aug 2, 2010 #1
    Boundary Value Problem + Green's Function
    Consider the BVP

    y''+4y=e^x
    y(0)=0
    y'(1)=0

    Find the Green's function for this problem.



    I am completely lost can someone help me out?
     
  2. jcsd
  3. Aug 2, 2010 #2
    I am no expert, this looks like just an ODE problem where you only have one independent variable x:

    1) Solve the associate homogeneous problem [itex]y''+4y=0[/itex] which is a second degree DE with constant coef.

    2) Then use variation of parameter to get the particular solution.

    3) [tex]y=y_c + y_p[/tex]

    4) Then use boundary condition to find the constant.

    Just my 2 cents
     
  4. Aug 2, 2010 #3
    yea but for a boundary condition should you have y(0) = c1 & y(t) = c2 , instead of y'(t) = c2
     
  5. Aug 3, 2010 #4
    I have not work through the problem, I am studying Green's function and that's the reason I look at this post.

    I don't see from [itex]y''+4y=e^x[/tex] that it is even multi variables. Only independent variable is x. From my understanding, Green's function only deal with multi-variables. This is a simple 2nd degree non-homogeneous ODE with constant coef. with boundary condition.
     
  6. Aug 3, 2010 #5
    Hi. I'm not very familiar with Green's functions, but let me suggest a direction I think you need to go:

    Consider the linear differential equation (of one variable):

    [tex]Ly=f[/tex]

    where the L is the differential operator like in your case, it's [itex]L=\frac{d^2}{dx^2}+4[/itex]. Then we can show the solution can be written in terms of a Green's function as:

    [tex]y(x)=\int G(x,u)f(u)du[/tex]

    where:

    [tex]G(x,u)=\sum_{n=1}^{\infty} \frac{\phi_n(x)\phi_n^{*}(u)}{\lambda_n}[/tex]

    where [itex]\phi_n[/itex] is a orthonormal set of eigenfunctions for the operator L, that is, normalized solutions to the equation:

    [tex]y''+4y=\lambda_n y[/tex]

    subject to the given boundary conditions. So, just need to find those huh? Also, keep in mind the conjugate ([itex]\phi^{*}[/itex]) of a real-valued function is just the function.

    See: "Mathematics of Classical and Quantum Physics" by F. Byron and R. Fuller. Whole chapter on Green's functions.
     
    Last edited: Aug 3, 2010
  7. Aug 3, 2010 #6
    I suggest taking a look at "Mathematical Physics" by Hassani or alternatively "Mathematical Methods for Physicists" by Arfken. Both have well developed sections on the use of Green's functions in solving ODEs.
     
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