Integral equations are equations in which an unknown function appears under an integral sign. Can we solve integral equations when the function is multivariable and the integral is a multiple integral?
For example, I want to find a function $f(x,y)$ that is a solution of the following integral equation.
$$ \int_0^y\int_0^x f(x',y') dx'dy' =(x-y)^2 $$
How can we find the $f(x,y)$?
We can solve some kinds of integral equations for a multivariable function. But in the example,
$$ \int_0^y\int_0^x f(x',y') dx'dy' =(x-y)^2 $$
is unsolvable. When $x=0$ or $y=0$ on the left side, the right side has to be always $0$, however, the left side is not $0$ for $x=0$ or $y=0$.