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Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial

Author

Listed:
  • Roberto Garrappa

    (Department of Mathematics, University of Bari, Via E. Orabona 4, 70126 Bari, Italy
    Member of the INdAM Research Group GNCS, Istituto Nazionale di Alta Matematica “Francesco Severi”, Piazzale Aldo Moro 5, 00185 Rome, Italy)

  • Eva Kaslik

    (Department of Mathematics and Computer Science, West University of Timisoara, Bd. V. Parvan 4, 300223 Timisoara, Romania)

  • Marina Popolizio

    (Member of the INdAM Research Group GNCS, Istituto Nazionale di Alta Matematica “Francesco Severi”, Piazzale Aldo Moro 5, 00185 Rome, Italy
    Department of Electrical and Information Engineering, Polytechnic University of Bari, Via E. Orabona 4, 70126 Bari, Italy)

Abstract

Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo’s derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications.

Suggested Citation

  • Roberto Garrappa & Eva Kaslik & Marina Popolizio, 2019. "Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial," Mathematics, MDPI, vol. 7(5), pages 1-21, May.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:407-:d:228974
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    References listed on IDEAS

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    1. Mariusz Ciesielski & Tomasz Blaszczyk, 2018. "An Exact Solution of the Second-Order Differential Equation with the Fractional/Generalised Boundary Conditions," Advances in Mathematical Physics, Hindawi, vol. 2018, pages 1-9, April.
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    Cited by:

    1. Zhang, Yufeng & Li, Jing & Zhu, Shaotao & Ma, Zerui, 2024. "Harmonic resonance and bifurcation of fractional Rayleigh oscillator with distributed time delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 221(C), pages 281-297.
    2. Abdelhamid Mohammed Djaouti & Zareen A. Khan & Muhammad Imran Liaqat & Ashraf Al-Quran, 2024. "A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives," Mathematics, MDPI, vol. 12(11), pages 1-20, May.
    3. Ruslan Abdulkadirov & Pavel Lyakhov & Nikolay Nagornov, 2023. "Survey of Optimization Algorithms in Modern Neural Networks," Mathematics, MDPI, vol. 11(11), pages 1-37, May.
    4. Enrica Pirozzi, 2022. "On a Fractional Stochastic Risk Model with a Random Initial Surplus and a Multi-Layer Strategy," Mathematics, MDPI, vol. 10(4), pages 1-18, February.
    5. Shiri, Babak & Baleanu, Dumitru, 2023. "All linear fractional derivatives with power functions’ convolution kernel and interpolation properties," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    6. Virginia Kiryakova, 2020. "Unified Approach to Fractional Calculus Images of Special Functions—A Survey," Mathematics, MDPI, vol. 8(12), pages 1-35, December.
    7. Virginia Kiryakova, 2021. "A Guide to Special Functions in Fractional Calculus," Mathematics, MDPI, vol. 9(1), pages 1-40, January.
    8. Daniele Mortari & Roberto Garrappa & Luigi Nicolò, 2023. "Theory of Functional Connections Extended to Fractional Operators," Mathematics, MDPI, vol. 11(7), pages 1-18, April.
    9. Ngueuteu Mbouna, S.G. & Banerjee, Tanmoy & Yamapi, René & Woafo, Paul, 2022. "Diverse chimera and symmetry-breaking patterns induced by fractional derivation effect in a network of Stuart-Landau oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).

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