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

Roberto Garrappa, Eva Kaslik and Marina Popolizio
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Roberto Garrappa: Department of Mathematics, University of Bari, Via E. Orabona 4, 70126 Bari, 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

Mathematics, 2019, vol. 7, issue 5, 1-21

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.

Keywords: fractional derivative; fractional integral; Mittag–Leffler function; Riemann–Liouville derivative; Caputo derivative; Grünwald–Letnikov derivative (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (9)

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