Fractional Calculus: A Comprehensive Review of Theory, Methods, and Applications

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@article{1717251,
      author = {Precious C. Agina, Elias I. Chukwuma, Kingsley K. Ibeh, Doris I. Ezeora},
      title = {Fractional Calculus: A Comprehensive Review of Theory, Methods, and Applications},
      journal = {Iconic Research And Engineering Journals},
      year = {2026},
      volume = {9},
      number = {11},
      pages = {242-252},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1717251.pdf},
      abstract = {Fractional calculus, which generalizes differentiation and integration to non-integer orders, has become an effective tool for modeling systems with memory and non-local characteristics. This paper provides a concise review of the fundamental theory, numerical methods, and key applications of fractional calculus. Core definitions, including the Riemann–Liouville and Caputo derivatives, are introduced alongside their main properties and physical interpretations. The review highlights widely used numerical techniques for solving fractional differential equations and discusses their computational challenges. Applications in physics, engineering, and related fields are examined, with particular emphasis on anomalous diffusion, viscoelastic systems, and fractional-order control. Comparisons with classical integer-order models demonstrate the enhanced modeling capability of fractional approaches in capturing complex dynamics. Finally, current challenges and research directions, including efficient computation and parameter identification, are outlined. This review aims to serve as a compact reference for researchers and practitioners working with fractional models.},
      keywords = {Fractional Calculus, Fractional Differential Equations, Memory Effects, Non-Local Dynamics, Numerical Methods.},
      month = {May},
      doi = {https://doi.org/10.64388/IREV9I11-1717251}
  }