In my research I encounter an eigenvalue integro-differential equation of the form:
$$f_n(x,y)=\lambda_n\iint_D\frac{\nabla'\cdot\big\lbrace h(x',y')\nabla'f_n(x',y')\big\rbrace}{\sqrt{(x-x')^2+(y-y')^2}}\,\mathrm{d}x'\mathrm{d}y',$$
where integration is over a 2-Dimensional area $D$ and $h(x,y)$ is a known function, and finally $\lambda_n$ and $f_n(x,y)$ are eigenvalues and eigenfunctions respectively . In this equation, primed versions of gradient operators indicate differentials with respect to primed coordinates (dummy variables of integration). I want to find eigenvalues and eigenfunctions of this equation numerically and analytically if possible; Anyone can help me,
Thank you in advance,