A new method of synthesis of the time-dependent behavior of a neutron pulse or of the space-dependent transfer function is given which combines analytical time dependence with numerical (or analytical) dependence of the other variables. The analytical development in Laguerre polynomials of time is obtained recursively in solving iteratively the static Boltzmann equations with sources. The convergence of the development, the bound of the truncation error, and the choice of the free parameters are examined for various cases. The method is finally applied (using an experimental map of the steady flux) in a water slab and the synthesized time-dependent flux is compared to the experimental one.