Modeling memory-dependent heat and momentum transport in fractional Jeffrey fluids under magnetic fields

Abstract Fractional-order formulations have proven to be particularly effective in viscoelastic fluid dynamics, as they capture memory and anomalous diffusion effects that are often missed by classical models, especially in non-Newtonian fluids such as Jeffrey fluids. In this work, a numerical framework is proposed to study a one-dimensional, coupled system of nonlinear fractional diffusion and energy equations governed by the Jeffrey fluid model using the Caputo fractional operator. The governing equations are first non-dimensionalized and then solved by employing the finite element method for spatial discretization and non-fractional terms, while the fractional derivatives are approximated using the L1 finite difference scheme. The novelty of this approach lies in the efficient hybrid treatment of the coupled nonlinear fractional system, supported by theoretical error estimates that confirm optimal convergence. Comprehensive graphical results are presented to illustrate solution behavior. MATLAB-based simulations demonstrate how variations in fractional order and key physical parameters influence velocity and temperature distributions, confirming the stability and accuracy of the proposed scheme. The results contribute to more reliable modeling of heat transfer in viscoelastic flows, with potential relevance to applications in polymer processing and biomedical fluid transport.