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\(\psi \)-Hilfer pseudo-fractional operator: new results about fractional calculus

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Abstract

In this paper, we introduce the \(\psi \)-Hilfer pseudo-fractional operator, motivated by the \(\psi \)-Hilfer fractional derivative and the theory of pseudo-analysis. We investigate a wide class of important and essential results for pseudo-fractional calculus in a semiring \(([a, b], \oplus , \odot )\) and some particular cases are discussed. Specifically, we present a class of pseudo-fractional operators which are particular cases of the \(\psi \)-Hilfer pseudo-fractional operator. In addition, we present the pseudo-Leibniz-type rules I and II and pseudo-Leibniz rules and some particular cases of Leibniz-type rules I and II are discussed. Finally, we obtain formulas for the Hilfer pseudo-fractional derivative, for the pseudo-Laplace transform, and for the g-integration by parts of the \(\psi \)-Hilfer pseudo-fractional operator.

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Acknowledgements

J. Vanterler acknowledges the financial support of a PNPD-CAPES (process number 873 no. 88882.305834/2018-01) scholarship of the Postgraduate Program in Applied Mathematics of IMECC874 Unicamp. The authors are grateful to Dr. J.E.M Maiorino for several and useful discussions and to the anonymous referees for suggestions that improved the manuscript.

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Authors and Affiliations

  1. Department of Applied Mathematics, Imecc-Unicamp, Campinas, SP, 13083-859, Brazil

    J. Vanterler da C. Sousa & E. Capelas de Oliveira

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

    Gastão S. F. Frederico

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  1. J. Vanterler da C. Sousa
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  2. Gastão S. F. Frederico
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  3. E. Capelas de Oliveira
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Correspondence to J. Vanterler da C. Sousa.

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Communicated by José Tenreiro Machado.

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Sousa, J.V.d.C., Frederico, G.S.F. & de Oliveira, E.C. \(\psi \)-Hilfer pseudo-fractional operator: new results about fractional calculus. Comp. Appl. Math. 39, 254 (2020). https://doi.org/10.1007/s40314-020-01304-6

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  • DOI: https://doi.org/10.1007/s40314-020-01304-6

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