Representation of Fractional Operators Using the Theory of Functional Connections

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 su...

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Bibliographic Details
Published inMathematics (Basel) Vol. 11; no. 23; p. 4772
Main Author Mortari, Daniele
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.12.2023
Subjects
Boundary conditions
Constraints
Derivatives
Fractional calculus
fractional derivative
fractional integral
Functional analysis
functional interpolation
Infinite series
Integers
Integrals
Methods
Mittag–Leffler function
Operator theory
Operators (mathematics)
Tests, problems and exercises
United States
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Summary: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.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math11234772