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

By Roberto Garrappa, Eva Kaslik, Marina Popolizio. Mathematics, 2019

112 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/e7c984e9b516cf7dcaada2734540143be5205486

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Citations in the corpus

Why fractional derivatives with nonsingular kernels should not be used (2020, background): “g functions are considered: fR(t) = ˆ 0 t2(1 ;0) f(t) t0 ; fC(t) = ˆ T m 1[y;0](t) t2(1 ;0) f(t) t0: The replacement of fby fR or fC, together with the property of the coecients !() j (see, e.g., [4]) that P 1 j=0 ! ( ) j j k= 0 for k= 0;1;:::;m 1, leads to the two distinct fractional derivatives GL”