Comments on various extensions of the Riemann-Liouville fractional derivatives: about the Leibniz and chain rule properties. (English) Zbl 1451.26008
Summary: Starting from the Riemann-Liouville derivative, many authors have built their own notion of fractional derivative in order to avoid some classical difficulties like a non zero derivative for a constant function or a rather complicated analogue of the Leibniz relation. Discussing in full generality the existence of such operator over continuous functions, we derive some obstruction Lemma which can be used to prove the triviality of some operators as long as the linearity and the Leibniz property are preserved. As an application, we discuss some properties of the Jumarie’s fractional derivative as well as the local fractional derivative. We also discuss the chain rule property in the same perspective.
MSC:
| 26A33 | Fractional derivatives and integrals |
| 49M25 | Discrete approximations in optimal control |
| 65Q30 | Numerical aspects of recurrence relations |
Keywords:
fractional derivations; Leibniz property; chain rule; obstruction lemma; Jumarie fractional derivative; local fractional derivativeReferences:
| [1] | Almeida, R.; Torres, D. F.M., Fractional variational calculus for nondifferentiable functions, Comput Math Appl, 61, 3097-3104 (2011) · Zbl 1222.49026 |
| [2] | Artstein-Avidan, S.; König, H.; Milman, V., The chain rule as a functional equation, J Funct Anal, 259, 2999-3024 (2010) · Zbl 1203.39014 |
| [3] | Adda, F. B.; Cresson, J., About non-differentiable functions, JMAA, 263, 721-737 (2001) · Zbl 0995.26006 |
| [4] | Adda, F. B.; Cresson, J., Fractional differential equations and the schrdinger equation, Appl Math Comput, 161, 1, 323-345 (2005) · Zbl 1085.34066 |
| [5] | Babakahani, A.; Daftardar-Gejji, V., On calculus of local fractional derivatives, JMAA, 270, 66-79 (2002) · Zbl 1005.26002 |
| [6] | Chen, Y.; Yan, Y.; Zhang, K., On the local fractional derivative, JMAA, 362, 17-33 (2010) · Zbl 1196.26011 |
| [7] | Cherbit, G., Dimension locale, quantit de mouvement et trajectoire, Fractals, dimension non entičre et applications, 340-352 (1967), Masson ed.: Masson ed. Paris |
| [8] | Cresson, J., Scale calculus and the Schrödinger equation, J Math Phys, 44, 12, 32 (2003) · Zbl 1062.39022 |
| [9] | Dias, N. C.; Prata, J. N., Local fractional derivatives, (Cresson, J., Fractional calculus in analysis, dynamics and optimal control (2014), Nova Sciences Publishers) |
| [10] | Giusti A. A comment on some new definitions of fractional derivative. Nonlinear Dyn 93(3):1757-1763. · Zbl 1398.28001 |
| [11] | Gonnord, S.; Tosel, N., Thémes d’analyse pour l’agrégation - calcul diffrentiel, Ed Ellipses (1998) |
| [12] | Hilfer R., Luchko Y. Desiderata for fractional derivatives and integrals. Mathematics 7(2):149. |
| [13] | Jacobson, N., Lie algebras (1962), Dover ed. · Zbl 0121.27504 |
| [14] | Jumarie, G., Modified Riemann-Liouville derivative and fractional taylor series of nondifferentiable functions, Further results, computers and mathematics with applications, vol. 51, 1367-1376 (2006) · Zbl 1137.65001 |
| [15] | Jumarie, G., Table of some basic fractional calculus formulae derived from a modified riemann-liouville derivative for non-differentiable functions, Appl Math Lett, 22, 378-385 (2009) · Zbl 1171.26305 |
| [16] | Jumarie, G., The leibniz rule for fractional derivatives holds with non-differentiable functions, Math Stat, 1, 2, 50-52 (2013) |
| [17] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations, North-Holland mathematics studies, vol. 204 (2006), Elsevier Science B.V.: Elsevier Science B.V. Amsterdam · Zbl 1092.45003 |
| [18] | Kolwankar, K. M.; Gangal, A. D., Fractional differentiability of nowhere differentiable functions and dimensions, Chaos, 6, 505-513 (1996) · Zbl 1055.26504 |
| [19] | Kolwankar, K. M.; Gangal, A. D., Local fractional derivatives and fractal functions of several variables, Proceedings of fractals in engineering (1997) |
| [20] | Kolwankar, K. M.; Véhel, J. L., Measuring functions smoothnes with local fractional derivatives, FCAA, 3, 4, 285-301 (2001) · Zbl 1044.26003 |
| [21] | König, H.; Milman, V., Characterizing the derivative and the entropy function by the Leibniz rule, J Funct Anal, 261, 1325-1344 (2011) · Zbl 1231.39011 |
| [22] | König, H.; Milman, V., Operator relations characterizing derivatives (2018), Birkhäuser · Zbl 1478.47001 |
| [23] | Liu, C.-s., Counterexamples on jumarie’s two basic fractional calculus formulae, Commun Nonlinear Sci Numer Simul, 22, 1-3, 92-94 (2015) · Zbl 1331.26010 |
| [24] | Liu, C.-s., On the local fractional derivative of everywhere non-differentiable continuous functions on intervals, Commun Nonlinear Sci NumerSimul, 42, 229-235 (2017) · Zbl 1473.26007 |
| [25] | Liu, C.-s., Counterexamples on jumarie’s three basic fractional calculus formulae for non-differentiable continuous functions, Chaos Solitons Fractals, 109, 219-222 (2018) · Zbl 1390.26011 |
| [26] | Ortigueira M.D., Machado J.T. What is a fractional derivative?JComput Phys 293:4-13. · Zbl 1349.26016 |
| [27] | Malinowska, A. B., Fractional variational calculus for non-differentiable functions, Fractional dynamics and control, 97-108 (2012), Springer: Springer New York |
| [28] | Ross, B.; Samko, S. G.; Love, E. R., Functions that have no first order derivative might have fractional derivatives of all orders less than one, Real Anal Exchange, 20, 2, 140-157 (1994/5) · Zbl 0820.26002 |
| [29] | Samko, S.; Kilbas, A.; Marichev, O., Fractional integrals and derivatives. Theory and applications (1993), Gordon and Breach · Zbl 0818.26003 |
| [30] | Tarasov, V. E., No violation of the leibniz rule. no fractional derivative, Commun Nonlinear Sci NumerSimul, 18, 11, 2945-2948 (2013) · Zbl 1329.26015 |
| [31] | Tarasov, V. E., On chain rules for fractional derivatives, Commun Nonlinear Sci Numer Simul, 30, 1-4 (2016) · Zbl 1489.26011 |
| [32] | Wang X. On the Leibniz rule and fractional derivatives for differentiable and non differentiable functions. 2014. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
