Numerical Simulation of a Time-Fractional Marriage-Divorce Model Using Bernstein Neural Networks

In this paper, we develop a time-fractional marriage–divorce (FMD) population model with four interacting compartments: singles, married, separated, and divorced. We model with Caputo fractional derivative to take into account long-term memory effects in social dynamics. We form a hybrid numerical framework by combining a BNN approximation with the classical L1 discretization scheme. Our numerical model has a high computational efficiency with vectorized implementation and it is still accurate with respect to the L1 scheme. In numerical experiments for fractional orders ξ ∈ (0; 1], we show that the proposed model is highly accurate and the error is small and bounded in the simulation domain. These results show that decreasing the fractional order increases memory effects, slows the transition dynamics, and significantly affects the evolution of all compartments. A parametric analysis shows that the key model parameters are important for the behavior of the system. Overall, the BNN-L1 framework is an efficient and reliable tool to solve nonlinear fractional compartmental models, which could also be applied to other social and applied dynamical systems.