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Pseudo-fractional operators of variable order and applications

  • Foundation, algebraic, and analytical methods in soft computing
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Abstract

The fractional calculus provides, over the decades, new tools based on formulations of definitions and discussions of properties, which allows greater connections with other areas. As highlighted, two pillars well-founded and built over these years, the first we highlight the fractional calculus that addresses integrals and derivatives of a non-variable order. Second, a natural consequence of the classic fractional calculus, they investigated the possibility of fractional integrals and derivatives of a variable order, although more restricted when discussing basic and fundamental properties of the fractional calculus. From these two pillars and the g-calculus theory (pseudo-analysis), a third pillar started to be built, although recently, but there are already some interesting results. In this sense, in the present paper, we present new extensions of pseudo-fractional operators for integral and derivative in the sense of g-calculus and investigate some essential properties of fractional calculus. In order to elucidate the results discussed, we present an application involving the \(\psi \)-pseudo fractional integral inequality of Chebyshev.

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Acknowledgements

Frederico acknowledges the financial support of the “Fundação Cearense de Apoio ao Desenvolvimento Científico Tecnológico” (FUNCAP) Agency Processo No. BP4-00172-00054.02.00/20.

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Authors and Affiliations

  1. Universidade Tecnológica Federal do Paraná, Guarapuava, PR, 85053-525, Brazil

    D. S. Oliveira

  2. Center for Mathematics, Computing and Cognition, Federal University of ABC, Avenida dos Estados, 5001, Bairro Bangu, SP 09.210-580, Santo André, Brazil

    J. Vanterler da C. Sousa

  3. Federal University of Ceará, Campus de Russas, Ceará, CE, 62900-000, Brazil

    Gastão S. F. Frederico

Authors
  1. D. S. Oliveira
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  2. J. Vanterler da C. Sousa
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  3. Gastão S. F. Frederico
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Correspondence to D. S. Oliveira.

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Oliveira, D.S., Sousa, J.V.d.C. & Frederico, G.S.F. Pseudo-fractional operators of variable order and applications. Soft Comput 26, 4587–4605 (2022). https://doi.org/10.1007/s00500-022-06945-9

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