Let $X = [a, b]^2$, and consider the unknown function $f:X\times [0, \infty) \to \mathbb{R}$. Also, let $A:X \to \mathbb{R}$ be a known function with positive entries. Assume $f$ satisfies the differential equation
\begin{equation*} \partial_tf(x, y, t) + A(x, y)f(x, y, t) - \int_{a}^{b}A(y, z)f(y, z, t)dz = 0. \end{equation*}
I've not run into something like this before, so I'm unsure if there are particular methods for finding $f$ in exact form, or if I'm going to ultimately have to reduce it to a computational problem. Any references would be welcome.