Results 1 to 3 of 3

Math Help - 3rd and Final Green Function Question

  1. #1
    Junior Member BrooketheChook's Avatar
    Joined
    Sep 2009
    From
    Gold Coast
    Posts
    27

    3rd and Final Green Function Question

    (Q4) Derive the Green's function for

     \displaystyle \frac{d^2y}{dx^2} - a^2y = \delta(x-x_0)

    with boundary conditions y(x)  \rightarrow 0 as x \pm \infty
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    First note that the solution is almost already is its self adjoint form.

    \frac{d}{dx}\left(-1\cdot y' \right)+ay=-\delta(x-x_0)

    So we need to find two different solutions such that each one satisfies one of the Boundary Conditions.

    v_0(t)=c_1e^{-ax}+c_2e^{ax}
    As x \to -\infty this gives c_1=0 and

    v_0(t)=c_2e^{ax}

    v_1(t)=c_3e^{-ax}+c_4e^{ax}

    As x \to \infty this gives c_4=0

    v_1(t)=c_3e^{-ax}

    So the Green's function has the form

    g(x,s)=\begin{cases} Ae^{as}\cdot e^{-ax} \text{ for } s \le x \\ Be^{ax}\cdot e^{-as} \text{ for } x \le s\end{cases}

    Now we use the Wronskian and the and the condiditon that

    p(x)W(x)=-1 so from above we have that p(x)=-1 and

    \begin{vmatrix} c_1e^{a(s-x)} && c_2e^{a(x-s)} \\ c_1(-a)e^{a(s-x)} &&  c_2(a)e^{a(x-s)} \end{vmatrix}=c_1c_2a+c_1_c2a

    Now using this we get

    2c_1c_2a(-1)=-1 \iff c_1c_2=\frac{1}{2a}

    Now we can pick c_1 and c_2 to be any real numbers that satisfy the above equation for example

    c_1=c_2=\frac{1}{\sqrt{2a}}

    This gives


    g(x,s)=\begin{cases} \frac{1}{\sqrt{2a}}e^{as}\cdot e^{-ax} \text{ for } s \le x \\ \frac{1}{\sqrt{2a}}e^{ax}\cdot e^{-as} \text{ for } x \le s\end{cases}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member BrooketheChook's Avatar
    Joined
    Sep 2009
    From
    Gold Coast
    Posts
    27
    Thank-you so much for this.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. One more Green's function question...Grrrr
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: November 7th 2010, 02:51 PM
  2. 2nd Green Function Question
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: November 5th 2010, 06:41 AM
  3. Question on normal derivative of Green's function.
    Posted in the Advanced Math Topics Forum
    Replies: 3
    Last Post: September 11th 2010, 08:44 AM
  4. Green's Function
    Posted in the Differential Equations Forum
    Replies: 6
    Last Post: April 10th 2010, 11:18 AM
  5. Green function
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 29th 2008, 12:00 AM

/mathhelpforum @mathhelpforum