Thread: Solving an integral equation

1. 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.