Application of Jumarie-Stancu Collocation Series Method and Multi-Step Generalized Differential Transform Method to fractional glucose-insulin

This study applies the Multi-Step Generalized Differential Transform Method (MSGDTM) and the Jumarie-Stancu Collocation Series Method (JSCSM) to analyze a fractional-order Model (1). The model incorporates Caputo fractional derivatives to capture the nonlocal and memory-dependent characteristics of glucose-insulin interactions, considering physiological factors such as β- cell activity and external glucose intake. Stability analysis reveals bifurcations and chaotic attractors, demonstrating the system’s sensitivity to fractional orders. Numerical simulations compare MSGDTM and JSCSM accuracy and efficiency, highlighting MSGDTM’s superior convergence and lower approximation error. The results show that fractional-order modeling provides a more accurate framework for understanding glucose-insulin dynamics and predicting metabolic behavior. Furthermore, control mechanisms are introduced to mitigate chaos, offering potential strategies for managing diabetes. This work emphasizes the robustness of MSGDTM in solving complex fractional biological models. It provides insights into fractional calculus applications in biomedical research.