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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.
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