This paper presents a novel fractional-order model for glucose-insulin dynamics using the Caputo-Fabrizio (CF) derivative, which accounts for memory effects through its nonsingular exponential kernel. Existence and uniqueness of the solution are established via fixed point theory, and infinite series solutions are obtained using the Sumudu transform. Hyers-Ulam stability is analyzed to assess the system’s robustness against perturbations. A linear control strategy is introduced to regulate glucose levels, demonstrating potential for integration with real-time insulin delivery systems. Compared to classical integer-order models, the proposed approach provides improved accuracy, enhanced stability, and deeper insight into the chaotic behavior of glucose-insulin interactions. This framework supports the development of personalized diabetes treatment and adaptive control strategies.

In this paper, we study the nonlinear weighted elliptic problem where B is the unit ball of , , and the singular logarithm weight with the limiting exponent in the Trudinger–Moser embedding. The nonlinearities are critical or subcritical growth in view of Trudinger–Moser inequalities. We prove the existence of nontrivial solutions via the critical point theory. In the critical case, the associated energy functional does not satisfy the compactness condition. We give a new growth condition and we point out its importance for checking the Palais–Smale compactness condition.

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.