Hilbert approximate solutions and fractional geometric behaviors of a dynamical fractional model of social media addiction affirmed by the fractional Caputo differential operator

•In this investigation, the dynamical model of social media addiction is considered in the spirit of the Caputo differential operator.•The main utilization is to elucidate the importance of nonclassical derivatives order in the implementation of social media addiction complex phenomena.•Firstly, we first developed the model using the Caputo scheme and discussed some fundamental mathematical computations.•Secondly, we adapt the Hilbert reproducing scheme to generate numerical appropriate solutions via Mathematica 12 software.•The convergence-error behavior results are established using functional analysis techniques.•A novel computational algorithm to handle social media addiction is utilized and discussed in detail.•The summary of the presented work and various scientific recommendations in addition to the future work that complements this analysis has been utilized in the last part. In this investigation, the dynamical model of social media addiction is considered in the spirit of the Caputo differential operator. The main utilization is to elucidate the importance of nonclassical derivatives order in the implementation of social media addiction complex phenomena. Herein, we first developed the model using the Caputo scheme and discussed some fundamental mathematical computations. Secondly, we adapt the Hilbert reproducing scheme to generate numerical appropriate solutions via Mathematica 12 software. The utilized fractional social media problem was formulated and configured to solve it by fitting two subspaces from the space of Hilbert. The convergence-error behavior results are established using functional analysis techniques. After utilizing several computational algorithms, the considered model results are tabularly and graphically demonstrated with different choices of arbitrary order parameter values. The outcomes show that when applying the fractional Caputo differential operator, such complicated events become more realistic and convincing. The summary of the presented work and various scientific recommendations in addition to the future work that complements this analysis has been utilized in the last part.