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An integral equation Abel and L operator?

  1. Apr 20, 2013 #1
    an integral equation Abel and L operator??

    1. The Abel operator
    The general Abel integral equation
    [tex]
    \begin{gathered}\intop_{x}^{a}\dfrac{F(y)dy}{\left(y^{2}-x^{2}\right)^{\frac{1+u}{2}}}=f(x)\end{gathered}
    [/tex]
    has the solution
    [tex]
    \begin{gathered}F(r)=-\dfrac{2\cos\frac{\pi u}{2}}{\pi}\dfrac{d}{dr}\intop_{r}^{a}\dfrac{f(x)xdx}{\left(x^{2}-r^{2}\right)^{\frac{1-u}{2}}}\end{gathered}
    [/tex]
    where [itex]\intop_{x}^{a}\frac{\left(\bullet\right)dy}{\left(y^{2}-x^{2}\right)^{\frac{1+u}{2}}}[/itex] is the well-known Abel operator
    2. The [itex]\mathcal{L}[/itex] operator
    The notation [itex]\lambda(k,\phi-\phi_{0})[/itex] is defined as follows:
    [tex]\begin{gathered}\lambda(k,\phi-\phi_{0}):=\dfrac{1-k^{2}}{1-2k\cos(\phi-\phi_{0})+k^{2}}\end{gathered}
    [/tex]
    The [itex]\mathcal{L}[/itex] operator is defined as
    [tex]\mathcal{L}(k)f(\phi)=\dfrac{1}{2\pi}\intop_{0}^{2\pi}\lambda(k,\phi-\phi_{0})f(\phi_{0})d\phi_{0}
    [/tex]
    Obviously, the following two properties for [itex] \mathcal{L} [/itex] operator are valid
    [tex]
    \begin{gathered}\mathcal{L}(k_{1})\mathcal{L}(k_{2})=\mathcal{L}(k_{1}k_{2})\\
    \lim_{k\rightarrow1}\mathcal{L}(k)f=f
    \end{gathered}
    [/tex]
    3. The question
    Prove the following integral equation
    [tex]
    \begin{gathered}4\intop_{0}^{\rho}\dfrac{dx}{\sqrt{\rho^{2}-x^{2}}}\end{gathered}
    \begin{gathered}\intop_{x}^{a}\dfrac{\rho_{0}d\rho_{0}}{\sqrt{\rho_{0}^{2}-x^{2}}}
    \mathcal{L}\left(\dfrac{x^{2}}{\rho\rho_{0}}\right)\sigma(\rho_{0,}\phi)=v(\rho_{0,}\phi)
    \end{gathered}
    [/tex]
    has the following solution
    [tex]
    \begin{gathered}\sigma(y,\phi)=\dfrac{1}{\pi^{2}}\left[\dfrac{\Phi(a,y,\phi)}{\sqrt{a^{2}-y^{2}}}-\intop_{y}^{a}\dfrac{dt}{\sqrt{t^{2}y^{2}}}\begin{gathered}\intop_{0}^{t}\dfrac{\rho d\rho}{\sqrt{t^{2}-\rho^{2}}}\mathcal{L}\left(\dfrac{\rho y}{t^{2}}\right)\Delta v(\rho_{,}\phi)\end{gathered}
    \right]\end{gathered}
    [/tex]
    where
    [tex]
    \begin{gathered}\Phi(t,y,\phi):=\dfrac{1}{t}\intop_{0}^{t}\dfrac{\rho d\rho}{\sqrt{t^{2}-\rho^{2}}}\dfrac{d}{d\rho}\left[\rho\mathcal{L}\left(\dfrac{\rho y}{t^{2}}\right)\Delta v(\rho_{,}\phi)\right]\end{gathered}
    [/tex]
    and [itex]\Delta[/itex] is the two-dimensional Laplace operator in polar coordinate, i.e.
    [tex]
    \begin{gathered}\Delta:=\dfrac{\partial^{2}}{\partial\rho^{2}}+\dfrac{1}{\rho^{2}}\dfrac{\partial^{2}}{\partial\phi^{2}}\end{gathered}
    [/tex]
    4.Hints
    To prove the formulae, one may make full use of the two properties of [itex] \mathcal{L} [/itex] operator and the approach of integration by parts.
    These are the hints in the book, but I still can not figure the final formulae out with the help of these hints. So I post it here and wait for your excellent proof.
     
  2. jcsd
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