The current numerical investigation illustrates how the fractional order parameter influences the dynamics of double-diffusive convective processes within a square porous enclosure featuring wavy walls. Specifically, the enclosure’s left (hot) and right (cold) wavy walls are kept at constant temperatures and solute concentrations, while the remaining two walls are thermally insulated and impermeable to solute. To analyze the transient behavior of fluid flow, the Caputo time-fractional derivative is applied to the energy and mass transfer equations, and momentum transport equation is modeled using the Darcy approach. In addition, the L1-scheme is employed to estimate the fractional time-derivative term, and the entire mathematical model is subsequently solved using the Galerkin finite element method. The investigation encompasses various parameters such as the Rayleigh number (Ra), buoyancy ratio (N), Lewis number (Le), and fractional-order parameter \((\alpha )\) . Moreover, simulations are carried out by varying the fractional-order parameter within the range \((0<\alpha \le 1)\) . The findings are presented through contour plots illustrating variations in isotherms, streamlines, and isoconcentrations, alongside numerical variations of the mean Nusselt number \((Nu_m)\) and mean Sherwood number \((Sh_m)\) . It is evident from the results that the fractional parameter \(\alpha \) significantly influences heat and solute transport phenomena, as well as the initial evolution states of streamlines, isotherms, and isoconcentrations.