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The Power Fractional Calculus: First Definitions and Properties with Applications to Power Fractional Differential Equations

Author

Listed:
  • El Mehdi Lotfi

    (Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University of Casablanca, Sidi Othman, Casablanca P.O. Box 7955, Morocco)

  • Houssine Zine

    (Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal)

  • Delfim F. M. Torres

    (Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal)

  • Noura Yousfi

    (Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University of Casablanca, Sidi Othman, Casablanca P.O. Box 7955, Morocco)

Abstract

Using the Laplace transform method and the convolution theorem, we introduce new and more general definitions for fractional operators with non-singular kernels, extending well-known concepts existing in the literature. The new operators are based on a generalization of the Mittag–Leffler function, characterized by the presence of a key parameter p . This power parameter p is important to enable researchers to choose an adequate notion of the derivative that properly represents the reality under study, to provide good mathematical models, and to predict future dynamic behaviors. The fundamental properties of the new operators are investigated and rigorously proved. As an application, we solve a Caputo and a Riemann–Liouville fractional differential equation.

Suggested Citation

  • El Mehdi Lotfi & Houssine Zine & Delfim F. M. Torres & Noura Yousfi, 2022. "The Power Fractional Calculus: First Definitions and Properties with Applications to Power Fractional Differential Equations," Mathematics, MDPI, vol. 10(19), pages 1-10, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:19:p:3594-:d:931371
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    References listed on IDEAS

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    1. Jianying Xiao & Yongtao Li, 2022. "Novel Synchronization Conditions for the Unified System of Multi-Dimension-Valued Neural Networks," Mathematics, MDPI, vol. 10(17), pages 1-24, August.
    2. Ding, Liang & Luo, Yi & Lin, Yan & Huang, Yirong, 2021. "Revisiting the relations between Hurst exponent and fractional differencing parameter for long memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 566(C).
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    Cited by:

    1. Xiaojun Zhou & Chunna Zhao & Yaqun Huang, 2023. "A Deep Learning Optimizer Based on Grünwald–Letnikov Fractional Order Definition," Mathematics, MDPI, vol. 11(2), pages 1-15, January.
    2. Ragwa S. E. Alatwi & Abdulrahman F. Aljohani & Abdelhalim Ebaid & Hind K. Al-Jeaid, 2022. "Two Analytical Techniques for Fractional Differential Equations with Harmonic Terms via the Riemann–Liouville Definition," Mathematics, MDPI, vol. 10(23), pages 1-11, December.

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