Fractional-order differential models for multiscale transport phenomena in materials and energy systems
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Shuwei Zhang
Tianjin Yongrui Business Information Consulting Co., Ltd., Tianjin 301600, China
Abstract
Transport in heterogeneous and hierarchical materials often exhibits memory, spatial nonlocality, distributed relaxation, and scale-dependent behavior that cannot be adequately represented by classical local constitutive laws. This review critically examines fractional-order differential models as effective continuum descriptions of multiscale heat, mass, charge, and reactive-species transport in materials and energy systems. It first reviews the principal classes of fractional formulations, including time-, space-, time-space-, variable-, distributed-, tempered-, and selectively fractionalized multiphysics models, and examines how operator definitions, kernel structures, domains, initial and boundary conditions, dimensional consistency, and stochastic or coarse-graining interpretations influence their physical meaning. The review then compares analytical, numerical, data-driven, and hybrid solution strategies while evaluating numerical verification, parameter identification, structural and practical identifiability, uncertainty quantification, model discrimination, and experimental validation. Applications in porous and heterogeneous materials, electrochemical systems, thermal transport, subsurface environments, reactive and degrading media, and engineered functional materials are assessed using three cumulative levels of evidence: phenomenological representation, mechanistic support, and predictive transferability. Classical models are retained as reference baselines throughout. The review shows that fractional models can provide compact descriptions of unresolved multiscale complexity; however, anomalous observations alone do not uniquely identify a fractional mechanism. Their scientific use is warranted only when the selected operator is physically and mathematically defensible, parameters are sufficiently identifiable, credible nonfractional alternatives are outperformed, and predictions remain transferable beyond calibration.
Copyright (c) 2026 Shuwei Zhang
This work is licensed under a Creative Commons Attribution 4.0 International License.
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