On chain rule for fractional derivatives. (English) Zbl 1489.26011
Summary: For some types of fractional derivatives, the chain rule is suggested in the form \(\mathcal{D}_x^\alpha f(g(x))= (\mathcal{D}_g^1 f(g))_{g=g(x)}\mathcal{D}_x^\alpha g(x)\). We prove that performing of this chain rule for fractional derivative \(\mathcal{D}_x^\alpha\) of order \(\alpha\) means that this derivative is differential operator of the first order \((\alpha=1)\). By proving three statements, we demonstrate that the modified Riemann-Liouville fractional derivatives cannot be considered as derivatives of non-integer order if the suggested chain rule holds.
MSC:
| 26A33 | Fractional derivatives and integrals |
References:
| [1] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach (1993), New York · Zbl 0818.26003 |
| [2] | Podlubny, I., Fractional differential equations (1998), Academic Press: Academic Press San Diego · Zbl 0912.34001 |
| [3] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003 |
| [4] | Ross, B., A brief history and exposition of the fundamental theory of fractional calculus, Fractional calculus and its applications: proceedings of the international conference held at the University of New Haven, 1-36 (1975), Springer-Verlag: Springer-Verlag Berlin · Zbl 0303.26004 |
| [5] | Debnath, L., A brief historical introduction to fractional calculus, Int J Math Educ Sci Technol, 35, 4, 487-501 (2004) |
| [6] | Machado, J. T.; Kiryakova, V.; Mainardi, F., Recent history of fractional calculus, Commun Nonlinear Sci Numer Simul, 16, 3, 1140-1153 (2011) · Zbl 1221.26002 |
| [7] | Machado, J. A.T; Galhano, A. M.; Trujillo, J. J., Science metrics on fractional calculus development since 1966, Fract Calc Appl Anal, 16, 2, 479-500 (2013) · Zbl 1312.26004 |
| [8] | Machado, J. A.T; Galhano, A. M.S. F.; Trujillo, J. J., On development of fractional calculus during the last fifty years, Scientometrics, 98, 1, 577-582 (2014) |
| [9] | Tarasov, V. E., Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media (2011), Springer: Springer New York |
| [10] | Encinas, L. H.; Masque, J. M., A short proof of the generalized Faá di Bruno’s formula, Appl Math Lett, 16, 6, 975-979 (2003) · Zbl 1041.26003 |
| [11] | Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput Math Appl, 51, 9-10, 1367-1376 (2006) · Zbl 1137.65001 |
| [12] | Jumarie, G., The Minkowski’s space-time is consistent with differential geometry of fractional order, Phys Lett A, 363, 1-2, 5-11 (2007) · Zbl 1197.83011 |
| [13] | Jumarie, G., Lagrangian mechanics of fractional order, Hamilton-Jacobi fractional PDE and Taylor’s series of nondifferentiable functions, Chaos Solitons Fractals, 32, 3, 969-987 (2007) · Zbl 1154.70011 |
| [14] | Jumarie, G., Oscillation of non-linear systems close to equilibrium position in the presence of coarse-graining in time and space, Nonlinear Anal Modell Control, 14, 2, 177-197 (2009) · Zbl 1196.93030 |
| [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, 3, 378-385 (2009) · Zbl 1171.26305 |
| [16] | Jumarie, G., From Lagrangian mechanics fractal in space to space fractal Schrödinger’s equation via fractional Taylor’s series, Chaos Solitons Fractals, 41, 1590-1604 (2009) · Zbl 1198.70019 |
| [17] | Jumarie, G., An approach via fractional analysis to non-linearity induced by coarse-graining in space, Nonlinear Anal Real World Appl, 11, 1, 535-546 (2010) · Zbl 1195.37054 |
| [18] | Jumarie, G., On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling, Open Phys, 11, 6, 617-633 (2013) |
| [19] | Tarasov, V. E., No violation of the Leibniz rule. No fractional derivative, Commun Nonlinear Sci Numer Simul, 18, 11, 2945-2948 (2013) · Zbl 1329.26015 |
| [20] | Tarasov, V. E., Comments on “The Minkowski’s space-time is consistent with differential geometry of fractional order”, [Physics Letters A 363 (2007) 5-11], Phys Lett A, 379, 14-15, 1071-1072 (2015) · Zbl 1341.83003 |
| [21] | Tarasov, V. E., Comments on Riemann-Christoffel tensor in differential geometry of fractional order application to fractal space-time [Fractals 21 (2013) 1350004], Fractals, 23, 2, 3p (2015) |
| [22] | Tarasov, V. E., Local fractional derivatives of differentiable functions are integer-order derivatives or zero, Int J Appl Comput Math, 1, 3 (2015) |
| [23] | Liu, Cheng-shi, Counterexamples on Jumarie’s two basic fractional calculus formulae, Commun Nonlinear Sci Numer Simul, 22, 1-3, 92-94 (2015) · Zbl 1331.26010 |
| [24] | Kolwankar, K. M.; Gangal, A. D., Fractional differentiability of nowhere differentiable functions and dimension, Chaos, 6, 4, 505-513 (1996) · Zbl 1055.26504 |
| [25] | Kolwankar, K. M.; Gangal, A. D., Hölder exponents of irregular signals and local fractional derivatives, Pramana J Phys, 48, 1, 49-68 (1997) |
| [26] | Kolwankar, K. M.; Gangal, A. D., Local fractional Fokker-Planck equation, Phys Rev Lett, 80, 2, 214-217 (1998) · Zbl 0945.82005 |
| [27] | Adda, F. B.; Cresson, J., About non-differentiable functions, J Math Anal Appl, 263, 2, 721-737 (2001) · Zbl 0995.26006 |
| [28] | Babakhani, A.; Daftardar-Gejji, V., On calculus of local fractional derivatives, J Math Anal Appl, 270, 1, 66-79 (2002) · Zbl 1005.26002 |
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