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A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

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Abstract

Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

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Authors and Affiliations

  1. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics, and Materials Hohai University, Nanjing, 210098, China

    HongGuang Sun, Ailian Chang & Wen Chen

  2. Department of Geological Sciences, University of Alabama, Tuscaloosa, AL, 35487, USA

    Yong Zhang

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  1. HongGuang Sun
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  2. Ailian Chang
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  3. Yong Zhang
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  4. Wen Chen
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Correspondence to HongGuang Sun.

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This paper won the “Riemann-Liouville Award: Best FDA Paper (Theory)” at the Conference ICFDA 2018.

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Sun, H., Chang, A., Zhang, Y. et al. A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications. FCAA 22, 27–59 (2019). https://doi.org/10.1515/fca-2019-0003

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