Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Help with Greens functions

  1. Jun 21, 2011 #1

    hunt_mat

    User Avatar
    Homework Helper

    I have to solve the following PDE:
    [tex]
    \frac{\partial^{2}u}{\partial t^{2}}+2\frac{\partial^{2}u}{\partial t\partial x}+\frac{\partial u}{\partial x}+\frac{\partial u}{\partial t}+k^{2}u=f
    [/tex]
    I use the Greens function method and examine the equation:
    [tex]
    \frac{\partial^{2}G}{\partial t^{2}}+2\frac{\partial^{2}G}{\partial t\partial x}+\frac{\partial G}{\partial x}+\frac{\partial G}{\partial t}+k^{2}G=4\pi\delta (x-x')\delta (t-t')
    [/tex]
    I then write:
    [tex]
    G=\frac{1}{2\pi}\int_{-\infty}^{\infty}g(X|X')e^{i\omega (T-T')}d\omega\quad \delta (T-T') =\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\omega (T-T')}d\omega
    [/tex]
    The equation then becomes:
    [tex]
    (1+2\omega i)\frac{\partial g}{\partial X}+(k^{2}-\omega^{2}+i\omega)g=4\pi\delta (X-X')
    [/tex]
    Take the Fourier transform to obtain:
    [tex]
    i\xi (1+2\omega i)\hat{g}+(k^{2}-\omega^{2}-\omega i)\hat{g}=4\pi e^{i\xi X'}
    [/tex]
    Rearrange and take the inverse Fourier transform to obtain:
    [tex]
    g=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{4\pi e^{-i\xi (X-X')}}{i\xi (1+2\omega i)+k^{2}-\omega^{2}-\omega i}d\xi
    [/tex]
    Am I on the right track here?
     
  2. jcsd
  3. Jun 21, 2011 #2

    hunt_mat

    User Avatar
    Homework Helper

    I think that I can find g by using contour integration. Write:
    [tex]
    \frac{1}{2\pi}\oint_{\gamma}\frac{4\pi e^{-iz(X-X')}}{iz(1+2\omega i)+k^{2}-\omega^{2}-\omega i}dz
    [/tex]
    Which can then be evaluated via Cauchy's integral formula:
    [tex]
    g=4\pi ie^{i(X-X')h(\omega )},\quad h(\omega )=\frac{k^{2}-\omega^{2}-\omega i}{1+2\omega i}
    [/tex]

    Thoughts?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Help with Greens functions
  1. Green's function (Replies: 3)

  2. Green's function (Replies: 6)

  3. Green functions (Replies: 3)

Loading...