Why can't the Kernel in the Volterra integral equation ever be separable? My textbook says it's obvious due to K(x,s)=0 at s>x, but I don't see how that helps to prove it.
u(x) = f(x) + \int_a^x K(x,s)x(s)ds
I tried to somehow use the fact that it can be rewritten as the Fredholm integral equation in a triangle area:
K(x,s) & a\leq s\leq x\\
0 & x\leq s\leq b
but so far no luck.
Assume separability. For the zeroing condition to hold, the s-dependent functions in the expansion of the kernel need to also depend on x (or how else do you make them zero?). This clearly contradicts separability.
I don't know what this means. You have K(x,s) in there... is that a typo? How is x an argument of K and also a function of s and also the argument of u and also a limit of integration?