1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Laplace transforms; Abel's integral equation

  1. Sep 19, 2008 #1
    Using Laplace transforms, find the solution of Abel's integral equation:

    [tex]\int^{x}_{0}\frac{f(u)}{\sqrt{x-u}}du = 1 + x + x^2 [/tex]

    I recognized that the integral is a Laplace convolution, leading to:

    [tex] (f*g)(x) = 1+x+x^2 [/tex]

    where [tex] g(x)=x^{-1/2}[/tex]


    [tex] L(f*g)=L(1)+L(x)+L(x^2)[/tex]

    [tex] L(f)L(g)=\frac{1}{p}+\frac{1}{p^2}+\frac{2}{p^3}[/tex]

    I can't figure out the transform of g(x). I tried contour integration in the first quadrant, indenting around the origin and placing the branch cut along the negative real axis, and I got to this:

    [tex] L(g)=\int^{\infty}_{0}\frac{e^{-px}}{\sqrt{x}}dx=i\int^{\infty}_{0}\frac{e^{-ipx}}{\sqrt{ix}}dx[/tex]

    Can anyone help me solve this last integral, or suggest another way to find the transform?
  2. jcsd
  3. Sep 19, 2008 #2
    Okay, I realized that I was making it too complicated by trying contour integration. I saw the change of variables x=y^2 for L(g), and got L(g)=sqrt(pi/p).

    So now I'm left with:

    [tex] L(f) = \sqrt{\frac{p}{\pi}}(\frac{1}{p}+\frac{1}{p^2}+\frac{2}{p^3}) [/tex]

    [tex] = \pi^{-1/2}(p^{-1/2}+p^{-3/2}+2p^{-5/2}) [/tex]

    Obviously I now know the inverse transform of the first term. The other two I can't get. I tried Bromwich contour integration, but the branch cut presents some difficulties there. Any ideas?
  4. Sep 19, 2008 #3
    I found the other transforms using some guesswork. I guess they can't be solved using a Bromwich contour, which to me has some interesting meta-mathematical implications.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Laplace transforms; Abel's integral equation