The fractional-order marriage-divorce mathematical model: numerical investigations and dynamical analysis

In this research, we present a fractional-order mathematical model that simulates the ongoing phenomenon of marriage and divorce, which significantly impacts human lives. The model is defined within the Liouville-Caputo framework and incorporates different states and parameter values. Specifically, the model consists of four state variables that describe the statuses of married, divorced, or troubled couples, as well as the interactions between these states. We conduct a stability analysis for the proposed model and calculate the equilibrium points, revealing the conditions for a stable solution. Moreover, we establish the existence, uniqueness, and boundedness of the model's solution using the Banach fixed point theorem. The global stability of the obtained equilibrium points is investigated using the Lyapunov function, revealing a globally stable solution with specific conditions. To obtain the results, we employ a multi-domain collocation technique that utilizes orthogonal generalized Romanovski polynomials. These polynomials offer the advantage of achieving higher-order accuracy without requiring a large number of basis functions. We assess the accuracy of the proposed technique by calculating the residual error. Furthermore, we conduct simulations with various parameter values and fractional orders to demonstrate the technique's effectiveness under different scenarios. The results confirm the technique's ability to provide accurate outcomes across multiple scenarios. To enhance result validation, a comparison with real data from the USA is introduced for both married and divorced individuals. This elucidates the significance of the proposed fractional model in providing a deeper understanding of the dynamics and behavior of the solution and its impact on the population's lives. By introducing and analyzing this model, this research contributes to a deeper understanding of the dynamics and implications of marriage and divorce phenomena.