Application of a fractal fractional operator to nonlinear glucose-insulin systems: Adomian decomposition solutions

This study proposes a generalized numerical scheme for simulating nonlinear fractal-fractional glucose-insulin systems with a Mittag-Leffler kernel. The model takes into account both the acute and chronic effects of beta-cell kinetic changes, and can be used to predict therapeutic interventions' effects. The model can also be used to identify potential targets for diabetes treatments. Solutions are examined for existence, uniqueness, non-negativity, and boundness. System stability is analyzed locally and globally, including Hyers-Ulam stability. Simulation results are compared to analytical solutions to validate the model. Finally, the effects of different parameters on the model are studied. We apply the Adomian decomposition method to approximate solutions to nonlinear glucose-insulin systems of fractional order when solving for fractional glucose levels. Based on this method, the solution can be represented as a series of easily computed convergent components. Numerical examples illustrate the method's effectiveness, showing that it accurately captures the system's dynamic behavior with precision. The generalized numerical scheme represents a significant advancement in glucose-insulin system simulation, providing a powerful and practical tool for researchers and clinicians. This tool can enhance the understanding and management of metabolic disorders, offering new insights into glucose and insulin interactions.