The neutron diffusion equation (NDE) is one of the most important partial differential equations (PDEs), to describe the neutron behavior in nuclear reactors and many physical phenomena. In this paper, we reformulate this problem via Caputo fractional derivative with integer-order initial conditions, whose physical meanings, in this case, are very evident by describing the whole-time domain of physical processing. The main aim of this work is to present the analytical exact solutions to the fractional neutron diffusion equation (F-NDE) with one delayed neutrons group using the Laplace transform (LT) in the sense of the Caputo operator. Moreover, the poles and residues of this problem are discussed and determined. To show the accuracy, efficiency, and applicability of our proposed technique, some numerical comparisons and graphical results for neutron flux simulations are given and tested at different values of time $t$ and order $\alpha$ which includes the exact solutions (when $\alpha =1).$ Finally, Mathematica software (Version 12) was used in this work to calculate the numerical quantities.