I want to find analytical solutions of the following integro-differential equation:
$\left(A\nabla_{\rho}^2 + B\nabla_z^2\right)f(\vec{r}) = C \int{ g(\vec{r},\vec{r\,'}) f(\vec{r\,'})d\vec{r\,'}}, \qquad g(\vec{r},\vec{r\,'}) = e^{-\alpha|z-z'|-\beta\rho^{\,'2}},$
where $A$, $B$, $C$, $\alpha$, and $\beta$ are real constants and $\nabla_{\rho}$ and $\nabla_z$ are gradients over $\rho$ and z, and $h$ is a well-known integrable function. I considered cylindrical coordinates where the position vector is given by $\vec{r} = \vec{\rho} + \vec{z}$ with $\vec{\rho} = \vec{x} + \vec{y}$.
It will be appreciated if you can leave your comments.
Best regards