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Hilfer–Katugampola fractional derivatives

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Abstract

We propose a new fractional derivative, the Hilfer–Katugampola fractional derivative. Motivated by the Hilfer derivative this formulation interpolates the well-known fractional derivatives of Hilfer, Hilfer–Hadamard, Riemann–Liouville, Hadamard, Caputo, Caputo–Hadamard, Liouville, Weyl, generalized and Caputo-type. As an application, we consider a nonlinear fractional differential equation with an initial condition using this new formulation. We show that this equation is equivalent to a Volterra integral equation and demonstrate the existence and uniqueness of solution to the nonlinear initial value problem.

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Acknowledgements

We would like to thank the anonymous referees and Dr. J. Emílio Maiorino for several suggestions and comments that helped improve the paper.

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Authors and Affiliations

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

    D. S. Oliveira & E. Capelas de Oliveira

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  1. D. S. Oliveira
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  2. E. Capelas de Oliveira
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Correspondence to D. S. Oliveira.

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Communicated by Eduardo Souza de Cursi.

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Oliveira, D.S., de Oliveira, E.C. Hilfer–Katugampola fractional derivatives. Comp. Appl. Math. 37, 3672–3690 (2018). https://doi.org/10.1007/s40314-017-0536-8

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  • DOI: https://doi.org/10.1007/s40314-017-0536-8

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