This work presents the development of an analytical approximation solution for a space-time-dependent neutron transport problem in a one-dimensional system consisting of a homogenized medium with a central external source with Green's functions. The delayed neutron production is implemented in two additional timescales with the multiple-scale expansion method. Qualitative results for a given system are analyzed, and a detailed comparison of the developed analytical approximation solution with results gained by the point-kinetics equation and the time-dependent diffusion equation without separation of space and time is given.