Application of Fractional-Order Modeling in Unsteady MHD Flow and Transport of Oldroyd-B Nanofluids in a Porous Medium Between Non-Isothermal Plates

This study presents a numerical investigation of unsteady magnetohydrodynamic (MHD) flow and heat–mass transfer in an Oldroyd-B nanofluid confined between non-isothermal parallel plates within a Darcy porous medium. The governing equations for velocity, temperature, and concentration incorporate distinct Caputo fractional-order time derivatives to model memory and hereditary effects. The model includes external magnetic and electric fields, Joule heating, Brownian motion, thermophoresis, and a first-order chemical reaction. A hybrid finite element–finite difference method is employed using linear basis functions and the L1 scheme for temporal discretization. A matrix-based framework solves the resulting nonlinear system, and error analysis with manufactured solutions confirms the method's accuracy and stability. Numerical results show that higher thermophoresis reduces the Nusselt and Sherwood numbers by 2.37% and 69.09%, respectively, while increasing (Formula presented.) enhances the Sherwood number by 40.10%. Skin friction declines with larger (Formula presented.). These findings have practical relevance in the design of electroconductive fluid systems, biomedical devices, and nanofluid-based thermal energy management in porous structures.