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Math Help - Solving an integral equation

  1. #1
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    Solving an integral equation

    1 The Abel operator
    The general Abel integral equation:
    \begin{gathered}\intop_{x}^{a}\dfrac{F(y)dy}{\left  (y^{2}-x^{2}\right)^{\frac{1+u}{2}}}=f(x)\end{gathered}
    has the solution
    \begin{gathered}F(r)=-\dfrac{2\cos\frac{\pi u}{2}}{\pi}\end{gathered}\dfrac{d}{dr}\intop_{r}^{  a}\dfrac{f(x)xdx}{\left(x^{2}r^{2}\right)^{\frac{1-u}{2}}}
    where \intop_{x}^{a}\frac{\left(\bullet\right)dy}{\left(  y^{2}-x^{2}\right)^{\frac{1+u}{2}}} is the well-known Abel operator
    2 The \mathcal{L} operator
    The notation  \lambda(k,\phi-\phi_{0}) is defined as follows:
    \begin{gathered}\lambda(k,\phi-\phi_{0}):=\dfrac{1-k^{2}}{1-2k\cos(\phi-\phi_{0})+k^{2}}\end{gathered}
    The \mathcal{L} operator is defined as:
    \mathcal{L}(k)f(\phi)=\dfrac{1}{2\pi}\intop_{0}^{2  \pi}\lambda(k,\phi-\phi_{0})f(\phi_{0})d\phi_{0}
    Obviously, the following two properties for \mathcal{L}operator are valid
    \mathcal{L}(k_{1})\mathcal{L}(k_{2})=\mathcal{L}(k  _{1}k_{2})
    \lim_{k\rightarrow1}\mathcal{L}(k)f=f
    3 The question
    Prove the following integral equation:
    \begin{gathered}4\intop_{0}^{\rho}\dfrac{dx}{\sqrt  {\rho^{2}-x^{2}}}\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}
    has the following solution
    \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}}}\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)\right]\end{gathered}
    where
    \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}
    and \Deltai the two-dimensional Laplace operator in polar coordinate, i.e.
    \begin{gathered}\Delta:=\dfrac{ \partial^{2}}{ \partial \rho^{2}}+\dfrac{1}{\rho^{2}}\dfrac{ \partial^{2}}{ \partial\phi^{2}}\end{gathered}
    4.Hints
    To prove the formulae, one may make full use of the two properties of \mathcal{L} 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.
    Last edited by jarvisyang; April 20th 2013 at 07:45 AM.
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