The one-dimensional tubular reactor model with advection and possibly axial diffusion is the classical model of distributed chemical reaction systems. This system is described by partial differential equations that depend on the time t and the spatial coordinate z. In this article, semi-analytical solutions to these partial differential equations are developed regardless of the complexity of their initial and boundary conditions and reaction kinetics. These semi-analytical solutions can be used to analyze the effect on the concentrations at the current coordinates z and t of (i) the initial and boundary conditions, and (ii) the reactions that took place at an earlier time. A case study illustrates the application of these results to tubular reactors for the two cases, without and with diffusion.