The flow and heat transfer characteristics of a tri-hybrid nanofluid in a porous medium are investigated under the influence of magnetohydrodynamics (MHD) and a ramped wall temperature. A Maxwell fluid is employed as the base fluid, in which three types of spherical nanoparticles, tungsten trioxide (WO₃), silver (Ag), and titanium dioxide (TiO₂), are suspended. The physical model is formulated using a system of partial differential equations subject to appropriate initial and boundary conditions. To enhance the novelty of the analysis, fractional derivatives are incorporated into the Maxwell fluid model along with porosity effects. Among the various definitions of fractional derivatives, the Caputo fractional derivative is preferred for its wide applicability in physical problems. The fractional-order derivatives are evaluated using the Caputo formulation, while the Crank–Nicolson numerical scheme is employed to discretize the time-dependent terms and solve the governing equations under ramped heating conditions. The proposed framework, which combines the Caputo fractional derivative with the Crank–Nicolson method to analyze tri-hybrid nanofluid flow, is a distinctive feature of this work. The Caputo derivative effectively captures memory-dependent behavior, allowing the model to account for the system’s dependence on its past states. This capability is particularly important for nanofluids exhibiting nonlocal and anomalous interactions, where classical integer-order models based on simple linear stress–strain relationships fail to accurately represent the complex rheological behavior. Overall, the adopted numerical approach provides improved accuracy and flexibility in modeling complex heat transfer processes, making the present study relevant to a wide range of biomedical and industrial applications.

The purpose of this study is to investigate the unsteady Caputo fractional fluid flow in the annular region of a vertical cylinder, incorporating heat supply or loss under natural convection and the influence of Hall current along a radial magnetic field. Fractional time derivatives of the Caputo type replace traditional integer-order derivatives to more accurately capture memory effects, anomalous diffusion, and sub-diffusive transport more accurately. To solve the resulting fractional model, a physics-informed neural network (PINN) framework is employed as a mesh-free alternative to conventional numerical techniques. By embedding the initial and boundary conditions, along with the governing fractional partial differential equations, directly into the loss function, the PINN effectively approximates both velocity and temperature fields. Automatic differentiation and the expressive capability of deep neural networks facilitate the treatment of concentric geometries and nonlocal fractional operators. The predicted results show strong agreement with existing literature, validating the accuracy of the proposed approach. Additionally, the results are computed and compared in terms of geometric aspects with small (λ=4) and large radial gaps (λ=10). Forλ=4, the curves are exactly parabolic, whereasλ=10, the profiles are attaining an asymptotic approach due to the ignorance of curvature effects for large r. The magnitude of the steady-state velocity at r = 4 increases by 10.29%, 52.33%, and 100% of its maximum velocity for the corresponding increment of α=0.1,0.5and0.9. Similarly, the temperature reaches 6.89%, 9.39%, and 100% of its maximum temperature for the same increments of α=0.1,0.5and0.9.