Hybrid Euler–Lagrange Approach for Fractional-Order Modeling of Glucose–Insulin Dynamics

We propose a hybrid Caputo–Lagrange Discretization Method (CLDM) for the fractional-order modeling of glucose–insulin dynamics. The model incorporates key physiological mechanisms such as glucose suppression, insulin activation, and delayed feedback with memory effects captured through Caputo derivatives. Analytical results establish positivity, boundedness, existence, uniqueness, and Hyers–Ulam stability. Numerical simulations confirm that the proposed method improves accuracy and efficiency compared with the Residual Power Series Method and the fractional Runge–Kutta method. Sensitivity analysis highlights fractional order θ as a biomarker for metabolic memory. The findings demonstrate that CLDM offers a robust and computationally efficient framework for biomedical modeling with potential applications in diabetes research and related physiological systems.