How to get from the first line to the second line? enter image description here
$$u_{n+1}(x,t) = u_n (x,t) - \int_0^t \left\{ (u_n)_s(x,s) - \frac{\partial}{\partial s} G(x,s) + \frac{\partial}{\partial s}E^{-1} \left\{ vE \left[ N u_n(x,s) \right] \right\} \right\}\, ds$$ Or alternatively $$ u_{n+1}(x,t) = G(x,t) - E^{-1} \left\{ vE \left[Nu_n (x,t) \right] \right\} $$