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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.

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# An integral equation Abel and L operator?

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