In the first part of this paper, we describe a review in the Laplace transform technique applied to SN two-dimensional transport equation in a rectangular domain considering Compton scattering. In Compton scattering the incident photon interacts with an electron of the irradiated medium, giving the electron kinetic energy, and also producing a scattered photon. In order to track that electron, in the second part of this paper, we apply the PN approximation to X,Y-geometry electron transport problems. In this derivation, the energy deposited by the free electron is taken into account. To reach this objective we solve two model problems: the first one for the transport of photons, assuming the Klein-Nishina scattering kernel as the scattering differential cross-section as well as the multigroup model in the energy variable; and the second one for the transport of free electrons considering the screened Rutherford scattering kernel. The results are presented under absorbed energy form in several points of interest. We report numerical simulations and comparisons with results obtained by using the Geant4 version 9.1 program which applies the Monte Carlo's technique to low energy libraries for both two-dimensional problems.