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Applications of Distributed-Order Fractional Operators: A Review.

Wei Ding1, Sansit Patnaik1, Sai Sidhardh1

  • 1Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA.

Entropy (Basel, Switzerland)
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Summary
This summary is machine-generated.

Distributed-order fractional calculus (DOFC) models complex systems by generalizing fractional operators. This review maps DOFC

Keywords:
control theorydistributed-order operatorsfractional calculustransport processesviscoelasticity

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Area of Science:

  • Fractional Calculus
  • Applied Mathematics
  • Complex Systems Modeling

Background:

  • Distributed-order fractional calculus (DOFC) is an advanced mathematical framework for modeling complex systems.
  • It extends constant and variable-order fractional operators to capture multiscale phenomena.
  • DOFC addresses systems with intricate nonlocal and memory effects across various scales.

Purpose of the Study:

  • To provide a systematic review of the state-of-the-art in distributed-order fractional calculus applications.
  • To offer a roadmap of DOFC's development and its application in real-world problem-solving.
  • To consolidate current research on DOFC for a broad audience.

Main Methods:

  • Review of mathematical foundations of DOFC, including analytical and numerical techniques.
  • Extensive survey of existing literature on DOFC applications.
  • Categorization of applications into key scientific and engineering domains.

Main Results:

  • DOFC offers significant opportunities for modeling complex systems with multiscale behaviors.
  • Key application areas include viscoelasticity, transport processes, and control theory.
  • A growing body of research highlights DOFC's utility in diverse scientific fields.

Conclusions:

  • DOFC is a powerful tool for understanding and modeling complex phenomena.
  • Further research is needed to fully explore its potential across various disciplines.
  • This review serves as a comprehensive guide to DOFC's current landscape and future directions.