Fractional-Order Advection–Diffusion Equations: A Critical Review of Physical Origins, Analytical and Numerical Methods, and Applications

Abstract

The advection–diffusion equation is the canonical continuum description of transport, yet it fails for a broad class of systems in which particle motion is non-Fickian. Fractional-order advection–diffusion equations replace integer derivatives with operators of non-integer order and, in doing so, encode long-range spatial jumps and long-memory waiting times that classical models discard. This review provides a critical, evidence-based synthesis of the field from its foundations in continuous-time random walks and Lévy motion through the mathematical and numerical landscape. We first examine the family of fractional operators and the physical meaning of their kernels. We then trace the micro–macro derivations that connect fractional advection–diffusion equations to anomalous transport, mobile–immobile systems, tempered and variable-order processes. Analytical methods and the now-mature toolbox of finite difference, finite element, spectral, meshless and lattice-Boltzmann discretizations are appraised for accuracy, stability and computational cost. Inverse problems and parameter estimation, which decide whether such models earn their extra complexity, are reviewed next. Applications in hydrology, porous media, plasmas, space physics, heat and mass transfer, biology and finance illustrate both success and unresolved tension. Throughout, we identify trends—rising use of non-singular-kernel operators, machine-learning solvers and realistic benchmarking—and we flag persistent gaps: inconsistent operator conventions, sparse validation against real data, and scarce higher-dimensional and multiscale results. We close with a concrete research agenda for making fractional advection–diffusion models predictive rather than descriptive. Keywords: fractional calculus; anomalous diffusion; advection–diffusion equation; Lévy motion; Caputo derivative; mobile–immobile transport; finite difference methods; inverse problems.