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Representation of Fractional Operators Using the Theory of Functional Connections

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  • Daniele Mortari

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

Abstract

This work considers fractional operators (derivatives and integrals) as surfaces f ( x , α ) subject to the function constraints defined by integer operators, which is a mandatory requirement of any fractional operator definition. In this respect, the problem can be seen as the problem of generating a surface constrained at some positive integer values of α for fractional derivatives and at some negative integer values for fractional integrals. This paper shows that by using the Theory of Functional Connections, all (past, present, and future) fractional operators can be approximated at a high level of accuracy by smooth surfaces and with no continuity issues. This practical approach provides a simple and unified tool to simulate nonlocal fractional operators that are usually defined by infinite series and/or complicated integrals.

Suggested Citation

  • Daniele Mortari, 2023. "Representation of Fractional Operators Using the Theory of Functional Connections," Mathematics, MDPI, vol. 11(23), pages 1-16, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:23:p:4772-:d:1288050
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    References listed on IDEAS

    as
    1. Daniele Mortari & Roberto Garrappa & Luigi Nicolò, 2023. "Theory of Functional Connections Extended to Fractional Operators," Mathematics, MDPI, vol. 11(7), pages 1-18, April.
    2. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
    3. Edmundo Capelas de Oliveira & José António Tenreiro Machado, 2014. "A Review of Definitions for Fractional Derivatives and Integral," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-6, June.
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