The Real-time Analysis for Particle-transport and In-situ Detection (RAPID) Code System, developed based on the Multi-stage Response-function Transport (MRT) methodology, enables real-time simulation of nuclear systems such as reactor cores, spent nuclear fuel pools and casks, and sub-critical facilities. This paper presents the application of a novel fission matrix-based burnup methodology to the well-characterized JSI TRIGA Mark II research reactor. This methodology allows for calculation of nuclear fuel depletion by combination and interpolation of RAPID's burnup dependent fission matrix (FM) coefficients to take into account core changes due to burnup. The methodology is compared to experimentally validated Serpent-2 Monte Carlo depletion calculations. The results show that the burnup methodology for RAPID (bRAPID) implemented into RAPID is capable of accurately calculating the keff burnup changes of the reactor core as the average discrepancies throughout the whole burnup interval are 37 pcm. Furthermore, capability of accurately describing 3D fission source distribution changes with burnup is demonstrated by having less than 1% relative discrepancies compared to Serpent-2. Good agreement is observed for axially and pin-wise dependent fuel burnup and nuclear fuel nuclide composition as a function of burnup. It is demonstrated that bRAPID accurately describes burnup in areas with high gradients of neutron flux (e.g. vicinity of control rods). Observed discrepancies for some isotopes are explained by analyzing the neutron spectrum. This paper presents a powerful depletion calculation tool that is capable of characterization of spent nuclear fuel on the fly while the reactor is in operation.