In this article we use one-dimensional, monoenergetic diffusion kinetic equation with one delayed neutron precursor concentration in Cartesian geometry to reconstruct the neutron flux of a reactor from the nuclear parameters, boundary, and initial conditions. The mathematical model is governed by a system of linear partial differential equations with prescribed boundary and initial conditions. As a matter of fact, the exact solution for any physical problem, if available, is of great importance which inevitably leads to a better understanding of the behavior of the involved physical phenomena. The present work represents an attempt for doing so, where the flux and precursor equations are solved by the help of Laplace transform in both spatial and time variables and consequently the exact expressions for the flux and concentration in space and time are established. We report numerical simulations as well study of numerical convergence of the obtained results.