Anomalous Diffusion in Drying: A Fractional Calculus Perspective on Modeling and Applications in Agro-Food Systems

Drying represents a critical operation in food, agricultural, and material processing systems; commonly modeled using Fick’s laws of diffusion. However, many real-world drying processes exhibit non-Fickian behavior, characterized by nonlinear moisture dynamics, internal resistance, and structural transformations, that classical models are unable to capture with sufficient accuracy.To address these limitations, fractional calculus has emerged as a robust mathematical framework for modeling anomalous diffusion; employing non-integer order derivatives that naturally incorporate memory effects and non-local transport phenomena. This review synthesizes recent advances in the application of fractional diffusion models to drying systems. Theoretical foundations of key fractional operators, including Caputo, Riemann-Liouville, and Grünwald-Letnikov, are examined, along with their physical interpretations in materials exhibiting structural heterogeneity, time-dependent moisture resistance, and coupled heat-mass transfer. Case studies involving fruits, vegetables, grains, hydrogels, and porous food matrices are analyzed; demonstrating the enhanced fitting and predictive capabilities of fractional models compared to classical approaches. Mathematical and computational techniques used to solve fractional differential equations are reviewed; encompassing analytical methods, finite difference and finite element schemes, and emerging tools such as physics-informed neural networks (PINNs). Challenges in parameter estimation are also addressed, with attention to the role of optimization algorithms and machine learning techniques in model calibration. By linking theoretical advancements with practical applications, this review highlights the increasing relevance of fractional diffusion frameworks in drying science; and outlines future directions in intelligent drying technologies and hybrid modeling strategies. • Fractional calculus provides a robust framework to model non-Fickian drying behavior • Fractional diffusion models capture memory effects and structural heterogeneity in foods • Case studies show superior fitting and predictive ability versus classical models • Numerical, analytical and PINN-based methods advance fractional drying simulations