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Theory of Functional Connections Extended to Fractional Operators

Author

Listed:
  • Daniele Mortari

    (Aerospace Engineering, Texas A&M University, College Station, TX 77845-3141, USA)

  • Roberto Garrappa

    (Department of Mathematics, Università degli Studi di Bari “Aldo Moro”, 70125 Bari, Italy
    GNCS Group, Istituto Nazionale di Alta Matematica (INdAM), 00185 Rome, Italy)

  • Luigi Nicolò

    (Department of Mathematics, Università degli Studi di Bari “Aldo Moro”, 70125 Bari, Italy)

Abstract

The theory of functional connections, an analytical framework generalizing interpolation, was extended and applied in the context of fractional-order operators (integrals and derivatives). The extension was performed and presented for univariate functions, with the aim of determining the whole set of functions satisfying some constraints expressed in terms of integrals and derivatives of non-integer order. The objective of these expressions was to solve fractional differential equations or other problems subject to fractional constraints. Although this work focused on the Riemann–Liouville definitions, the method is, however, more general, and it can be applied with different definitions of fractional operators just by changing the way they are computed. Three examples are provided showing, step by step, how to apply this extension for: (1) one constraint in terms of a fractional derivative, (2) three constraints (a function, a fractional derivative, and an integral), and (3) two constraints expressed in terms of linear combinations of fractional derivatives and integrals.

Suggested Citation

  • Daniele Mortari & Roberto Garrappa & Luigi Nicolò, 2023. "Theory of Functional Connections Extended to Fractional Operators," Mathematics, MDPI, vol. 11(7), pages 1-18, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1721-:d:1115564
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    References listed on IDEAS

    as
    1. Roberto Garrappa, 2018. "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial," Mathematics, MDPI, vol. 6(2), pages 1-23, January.
    2. Ahmad, Wajdi M. & El-Khazali, Reyad, 2007. "Fractional-order dynamical models of love," Chaos, Solitons & Fractals, Elsevier, vol. 33(4), pages 1367-1375.
    3. Daniele Mortari, 2017. "The Theory of Connections: Connecting Points," Mathematics, MDPI, vol. 5(4), pages 1-15, November.
    4. 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.
    5. Tarasov, Vasily E., 2020. "Fractional econophysics: Market price dynamics with memory effects," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
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    Cited by:

    1. Daniele Mortari, 2023. "Representation of Fractional Operators Using the Theory of Functional Connections," Mathematics, MDPI, vol. 11(23), pages 1-16, November.

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