Novel application of q-HAGTM to analyze Hilfer fractional differential equations in diabetic dynamics

Fractional calculus has emerged as a valuable tool for modeling complex dynamical systems due to its ability to represent the memory and hereditary characteristics. Among the various fractional operators, the Hilfer fractional derivative is particularly notable for its unique capability to interpolate between the Caputo and Riemann–Liouville derivatives, providing a versatile framework for fractional-order modeling. Although various numerical and analytical methods have been used to solve Hilfer fractional differential equations, the q-Homotopy analysis generalized transform method remains unexplored. In this study, we extend a glucose–insulin interaction model to a Hilfer fractional framework and apply this method to analyze its dynamics. The approach provides an effective solution technique, highlighting its potential for analyzing complex fractional-order biological systems. To establish the validity of our approach, we analyze the uniqueness and convergence of the obtained solutions. Numerical simulations and graphical representations illustrate the model’s behavior and confirm the method’s effectiveness. This study presents a novel application of the q-Homotopy analysis generalized transform method to Hilfer fractional equations. The results confirm the applicability and effectiveness of this method in obtaining reliable solutions. The findings contribute to the broader understanding of fractional-order modeling and demonstrate the potential of this approach for future research in mathematical modeling.