Generalized fractional operators do not preserve periodicity

Roberto Garrappa, Katarzyna Górska, Eva Kaslik, Kateryna Marynets

Fractional Calculus and Applied Analysis (FCAA)

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Abstract

This work allows proving that the action of fractional derivatives and fractional integrals on periodic functions does not preserve the periodicity of any period. This result is proved not only for one type of fractional operator but also for the wide class of generalized fractional operators based on the Sonine condition, a class that encompasses the majority of the fractional operators commonly used. Moreover, for several specific fractional operators, we provide explicit representations of the derivatives and integrals of the sine function, showing that they are composed of a local periodic term and a non-local term, which is the cause of the loss of periodicity.

BibTeX

@article{GarrappaGorskaKaslikMarynets2025,
  title         = {Generalized Fractional Operators Do Not Preserve Periodicity},
  author        = {
    Roberto Garrappa and Katarzyna G\'{o}rska and Eva Kaslik and Kateryna
    Marynets
  },
  year          = 2025,
  month         = jun,
  day           = 4,
  journal       = {Fractional Calculus and Applied Analysis},
  publisher     = {Springer Science and Business Media {LLC}},
  doi           = {10.1007/s13540-025-00427-z},
  issn          = {1314-2224},
  url           = {https://doi.org/10.1007/s13540-025-00427-z},
  language      = {en},
  abstract      = {
    This work allows proving that the action of fractional derivatives and
    fractional integrals on periodic functions does not preserve the
    periodicity of any period. This result is proved not only for one type of
    fractional operator but also for the wide class of generalized fractional
    operators based on the Sonine condition, a class that encompasses the
    majority of the fractional operators commonly used. Moreover, for several
    specific fractional operators, we provide explicit representations of the
    derivatives and integrals of the sine function, showing that they are
    composed of a local periodic term and a non-local term, which is the cause
    of the loss of periodicity.
  },
}