Desiderata for Fractional Derivatives and Integrals
Abstract
:- (a)
- Integrals and derivatives of fractional order should be linear operators on linear spaces3.
- (b)
- On some subset4 the index law (semigroup property)
- (c)
- Restricted to a suitable subset of the domain of the fractional derivatives of order operate as left inverses
- (d)
- There is a subset of the domain of such that the limits
- (e)
- The limiting map is the identity on , i.e., ;
- (f)
- The limiting map is a derivation on . This means it is possible to define a multiplication on such that the Leibniz rule
Appendix A
- (a)
- .
- (b)
- for all .
- (c)
- For every the orbit maps are continuous from into .
References
- Ross, B. A brief history and exposition of the fundamental theory of fractional calculus. In Fractional Calculus and its Applications; Ross, B., Ed.; Springer Verlag: Berlin, Germany, 1975; Volume 457, pp. 1–37. [Google Scholar]
- Ortigueira, M.; Tenreiro-Machado, J. What is a fractional derivative? J. Comput. Phys. 2015, 293, 4–13. [Google Scholar] [CrossRef]
- Hilfer, R. Thermodynamic Scaling Derived via Analytic Continuation from the Classification of Ehrenfest. Phys. Scr. 1991, 44, 321. [Google Scholar] [CrossRef]
- Hilfer, R. Multiscaling and the Classification of Continuous Phase Transitions. Phys. Rev. Lett. 1992, 68, 190. [Google Scholar] [CrossRef] [PubMed]
- Hilfer, R. Fractional Calculus and Regular Variation in Thermodynamics. In Applications of Fractional Calculus in Physics; Hilfer, R., Ed.; World Scientific: Singapore, 2000; p. 429. [Google Scholar]
- Hilfer, R. Mathematical and physical interpretations of fractional derivatives and integrals. In Handbook of Fractional Calculus and Applications, Volume 1: Basic Theory; Kochubei, A., Luchko, Y., Eds.; De Gruyter: Berlin, Germany, 2019; p. 47. [Google Scholar]
- Liouville, J. Mémoire sur quelques Questions de Geometrie et de Mecanique, et sur un nouveau genre de Calcul pour resoudre ces Questions. J. l’Ecole Polytech. 1832, XIII, 1. [Google Scholar]
- Kochubei, A.; Luchko, Y. Basic FC operators and their properties. In Handbook of Fractional Calculus and Applications, Volume 1: Basic Theory; Kochubei, A., Luchko, Y., Eds.; De Gruyter: Berlin, Germany, 2019; p. 23. [Google Scholar]
1 | properties to be desired. |
2 | It is common to use only one of the symbols or in the sense that either or . In this paper we keep the distinction between and by assuming unless otherwise specified. This entails discussing the case separately whenever necessary. |
3 | Dependencies of and on other parameters are usually present, but notationally suppressed. |
4 | Here the index (b) refers to desideratum (b). The same applies in desiderata (d)–(f) below. |
5 | (a) for all also , (b) , (c) , (d) there exists an element (called origin ) such that for all , (e) for all there is an element such that (f) for all (or ) and an element is defined, (g) for all (or ) and one has , (h) for all (or ) and one has , (i) for all (or ) and one has , and (j) for all . |
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Hilfer, R.; Luchko, Y. Desiderata for Fractional Derivatives and Integrals. Mathematics 2019, 7, 149. https://doi.org/10.3390/math7020149
Hilfer R, Luchko Y. Desiderata for Fractional Derivatives and Integrals. Mathematics. 2019; 7(2):149. https://doi.org/10.3390/math7020149
Chicago/Turabian StyleHilfer, Rudolf, and Yuri Luchko. 2019. "Desiderata for Fractional Derivatives and Integrals" Mathematics 7, no. 2: 149. https://doi.org/10.3390/math7020149
APA StyleHilfer, R., & Luchko, Y. (2019). Desiderata for Fractional Derivatives and Integrals. Mathematics, 7(2), 149. https://doi.org/10.3390/math7020149