I came across Volterra integral equations in a completely different context: branching theory applied to macroevolution. Wikipedia mentions renewal theory, which is close. In my application, the equation could be solved iteratively. I think this notation
$ Q(\omega) \ast F(\omega) $
is confusing and will use
$ (Q \ast F)(\omega) $.
The idea is to write
$ F(\omega) = (\omega^2 + a)^{-1} [ H(\omega) - (Q \ast F)(\omega) ], $
take a guess at $F(\omega)$, say $F_0(\omega)=0$, and iterate like
$ F_{i+1}(\omega) = (\omega^2 + a)^{-1} [ H(\omega) - (Q \ast F_i)(\omega) ] .$
Numerically, the $F_i$s are evaluated at a finite number of evenly spaced $\omega$ values. In my case this procedure was guaranteed to converge.
$ Q(\omega) \ast F(\omega) $
is confusing and will use
$ (Q \ast F)(\omega) $.
The idea is to write
$ F(\omega) = (\omega^2 + a)^{-1} [ H(\omega) - (Q \ast F)(\omega) ], $
take a guess at $F(\omega)$, say $F_0(\omega)=0$, and iterate like
$ F_{i+1}(\omega) = (\omega^2 + a)^{-1} [ H(\omega) - (Q \ast F_i)(\omega) ] .$
Numerically, the $F_i$s are evaluated at a finite number of evenly spaced $\omega$ values. In my case this procedure was guaranteed to converge.
Version 2.1.8p2
