Abstract
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 \(L_2\) 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 \(\theta _c\) 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 (\(\alpha =0.9\), \(\beta =0.5\)) 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.
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The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.
Abbreviations
- \(\alpha\) :
-
Order of the time-fractional Caputo derivative (dimensionless)
- \(\beta\) :
-
Order of the space-fractional Riesz derivative (dimensionless)
- \(\theta\) :
-
Time delay in the coupled system (s)
- \(\lambda\) :
-
Characteristic eigenvalue (\(\hbox {m}^{-1}\))
- \(\omega\) :
-
Oscillation frequency (\(\hbox {rad}\cdot \hbox {s}^{-1}\))
- \(\Omega\) :
-
Bounded spatial domain (m)
- \(\kappa\) :
-
Thermal diffusivity/diffusion coefficient (\(\hbox {m}^2\cdot \hbox {s}^{-1}\))
- \(\mu\) :
-
Damping coefficient (\(\hbox {N}\cdot \hbox {s}\cdot \hbox {m}^{-1}\))
- \(\nu\) :
-
Coupling strength between oscillator and diffusion (dimensionless)
- \(D_t^\alpha\) :
-
Caputo time-fractional derivative of order \(\alpha\)
- \(D_x^\beta\) :
-
Riesz space-fractional derivative of order \(\beta\)
- \(I_t^\alpha\) :
-
Riemann–Liouville fractional integral of order \(\alpha\)
- u(t, x):
-
Displacement/temperature field (m / K)
- t :
-
Time (s)
- x :
-
Spatial coordinate (m)
- \(L^2(\Omega )\) :
-
Space of square-integrable functions on \(\Omega\)
- \(H^\beta (\Omega )\) :
-
Fractional Sobolev space of order \(\beta\)
- \(H_0^1(\Omega )\) :
-
Sobolev space with homogeneous Dirichlet boundary conditions
- \(\Vert \cdot \Vert _{L^2}\) :
-
\(L^2\)-norm
- \(\Vert \cdot \Vert _X\) :
-
Norm in the Banach space \(X = C([0,T]; L^2(\Omega ))\)
- \(C_{\alpha ,\Omega }\) :
-
Embedding constant depending on \(\alpha\) and domain \(\Omega\)
- \(C_{\alpha ,\beta }\) :
-
Interpolation constant depending on \(\alpha\) and \(\beta\)
- L :
-
Contraction constant for the fixed-point operator (dimensionless)
- T :
-
Final time of simulation (s)
- k :
-
Thermo-mechanical coupling coefficient (\(\hbox {W}\cdot \hbox {sm}^{-1}\cdot \hbox {K}^{-1}\))
- f(t, x):
-
External source/loading term (MPa / \(\hbox {W}\cdot \hbox {m}^{-3}\))
- \(\Delta t\) :
-
Time step for numerical discretization (s)
- \(\Delta x\) :
-
Spatial grid size (m)
- FDM:
-
Finite Difference Method
- GL:
-
Grünwald–Letnikov
- PDE:
-
Partial Differential Equation
- FDE:
-
Fractional Differential Equation
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Funding
This work is partially supported by the Support Plan on Science and Technology for Youth Innovation of Universities in Shandong Province (Grant Nos. 2021KJ086).
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T.X. Li conceived the research framework, formulated the fractional-order coupled model, and led the theoretical analysis. Y.F. Zhang conducted the mathematical proofs for the existence, uniqueness, and stability of solutions, and derived the key theorems. X.D. Zhao performed the numerical simulations, analyzed the parameter influences, and visualized the results. Y. Wang and Y.Q. Hu investigated the time-delay induced bifurcation characteristics, derived the critical conditions and stability criteria, and contributed to the discussion section. All authors reviewed the manuscript, refined the theoretical arguments, and approved the final version.
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Li, T., Zhao, X., Zhang, Y. et al. A novel fractional-order coupled model integrating a damped oscillator equation with a non-Fickian heat conduction equation. Sci Rep (2026). https://doi.org/10.1038/s41598-026-44718-8
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DOI: https://doi.org/10.1038/s41598-026-44718-8
