Numerical Analysis of Fractal-Fractional Modelling of a Glucose-Insulin Model with Power-Law Kernel via Newton Interpolation: Application of the Laplace-Adomian Decomposition method

In this paper, a power-law kernel is incorporated into the Caputo operator framework in order to construct a novel fractal-fractional model of the glucose-insulin regulatory system. Memory effects and complex physiological dynamics associated with glucose-insulin interactions are captured in the model. In order to obtain approximate analytical solutions, we use the Laplace-Adomian Decomposition Method (LADM) combined with the Newton interpolation polynomial. A theoretical analysis validates the proposed scheme via fixed-point theory, as well as Lyapunov-based criteria for boundedness and non-negativity. Furthermore, the influence of fractional parameters on insulin and glucose dynamics is investigated. We conducted numerical simulations to evaluate the model's behavior under various fractional orders. The results demonstrate that the model exhibits a slower glucose decay and a delayed insulin response as the fractional order decreases, which is in good agreement with experimental observations. The LADM-based approach is shown to be effective in solving nonlinear fractional biological systems, providing insight into the role of memory and hereditary properties. The quantitative performance of the method is also assessed by computing L2 error norms and convergence orders, confirming its accuracy and efficiency for biomedical modeling.