A novel fractional-order coupled model integrating a damped oscillator equation with a non-Fickian heat conduction equation.

This paper presents a novel fractional-order coupled model that integrates a damped oscillator equation with a non-Fickian heat conduction equation, tailored to characterize the thermo-mechanical behavior of nanohybrid materials. The model employs time-fractional derivatives to capture the memory effects in viscoelastic damping arising from nanofiller-matrix interactions, while space-fractional derivatives describe anomalous heat transport in hierarchical microstructures. A rigorous theoretical framework is established: the existence and uniqueness of solutions are proven via the Banach fixed-point theorem, and uniform stability in the [Formula: see text] sense is demonstrated using an energy function method. Furthermore, the dynamic behavior of the system with time delay is systematically investigated, deriving explicit criteria for Hopf bifurcation, including the critical time delay [Formula: see text] and the conditions for supercritical or subcritical bifurcation. Numerical simulations, using the L1 and Grünwald-Letnikov schemes, are conducted to compare the fractional-order model ([Formula: see text], [Formula: see text]) with its integer-order counterpart. The results show that the fractional model preserves more pronounced memory properties and exhibits a slower decay over extended time scales, which is crucial for depicting the long-term dynamic and diffusion characteristics of nanohybrid materials. The proposed framework not only enriches the theoretical system of fractional-order coupled differential equations but also provides a reliable mathematical tool for the dynamic analysis and stability control of thermo-mechanical systems in engineering applications, such as nanocomposite materials and aerospace structures.