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

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 pri...

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Bibliographic Details
Published inFractional calculus & applied analysis Vol. 22; no. 1; pp. 27 - 59
Main Authors Sun, HongGuang, Chang, Ailian, Zhang, Yong, Chen, Wen
Format Journal Article
LanguageEnglish
Published Warsaw Versita 01.02.2019
De Gruyter
Nature Publishing Group
Subjects
34A34
34A45
34K28
35R11
60G22
65M12
Abstract Harmonic Analysis
Analysis
applications
Computer simulation
Dependent variables
Differential equations
fractional calculus
fractional differential equations
Functional Analysis
Integral Transforms
Mathematical models
Mathematics
Numerical analysis
Numerical methods
Operational Calculus
Primary 26A33
Secondary 34A08
Survey Paper
Time dependence
variable-order
60G22
variable-order
fractional calculus
numerical methods
Primary 26A33
34A34
34A45
fractional differential equations
Secondary 34A08
34K28
35R11
65M12
applications
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Summary: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.
Bibliography:ObjectType-Article-1
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ISSN:1311-0454
1314-2224
DOI:10.1515/fca-2019-0003