Fractional Calculus in Physics: A Brief Review of Fundamental Formalisms

Cresus Fonseca de Lima Godinho and Ion Vasile Vancea ()
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Cresus Fonseca de Lima Godinho: Group of Theoretical and Mathematical Physics, Department of Physics, Federal Rural University of Rio de Janeiro, Cx. Postal 23851, BR 465 Km 7, Seropédica 23890-000, RJ, Brazil
Ion Vasile Vancea: Group of Theoretical and Mathematical Physics, Department of Physics, Federal Rural University of Rio de Janeiro, Cx. Postal 23851, BR 465 Km 7, Seropédica 23890-000, RJ, Brazil

Mathematics, 2025, vol. 13, issue 22, 1-22

Abstract: Fractional calculus provides powerful tools for modeling nonlocality, dissipative systems, and, when defined in the time representation, provides an interesting memory effect in mathematical physics. In this paper, we review four standard fractional approaches: the Riemann–Liouville, Gerasimov–Caputo, Grünwald–Letnikov, and Riesz formulations. We present their definitions, basic properties, Weyl–Marchaud, and physical interpretations. We also give a brief review of related operators that have been used recently in applications but have received less attention in the physical literature: the fractional Laplacian, conformable derivatives, and the Fractional Action-Like Variational Approach (FALVA) for variational principles with fractional action weights. Our emphasis is on how these operators are, and can be, applied in physical problems rather than on exhaustive coverage of the field. This review is intended as an accessible introduction for physicists working in diverse areas interested in fractional calculus and fractional methods. For deeper technical or domain-specific treatments, readers are encouraged to consult the works in the corresponding fields, for which the bibliography suggests a starting point.

Keywords: fractional calculus; fractional derivative; fractional integral; non-local operators; conformable derivatives; fractional Laplacian; FALVA (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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