Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials

TL;DR

This survey reviews fractional-order differential equations as effective models for complex phenomena like subsurface transport, turbulence, and anomalous materials, highlighting their advantages and future research directions.

Contribution

It provides a comprehensive overview of fractional derivatives, surveys key application areas, and discusses future research avenues for more advanced models and calibration tools.

Findings

Fractional models effectively describe anomalous transport phenomena.

Evidence supports the use of fractional derivatives in turbulence and materials.

Future research includes developing physically sound models and calibration methods.

Abstract

Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, due to the inherent ability to describe large-scale behavior with greater efficiency than fully-resolved classical models. In this review article, we first provide a broad overview of fractional-order derivatives with a clear emphasis on the stochastic processes that underlie their use. We then survey three exemplary application areas - subsurface transport, turbulence, and anomalous materials - in which fractional-order differential equations provide accurate and predictive models. For each area, we report on the evidence of anomalous behavior that justifies the use of fractional-order models, and survey both foundational models as well as more expressive state-of-the-art models. We also propose avenues for future research, including more advanced and physically sound models, as well as tools for calibration and discovery of fractional-order models.