0
Research Papers

Leibniz Rule and Fractional Derivatives of Power Functions

[+] Author and Article Information
Vasily E. Tarasov

Skobeltsyn Institute of Nuclear Physics,
Lomonosov Moscow State University,
Moscow 119991, Russia
e-mail: tarasov@theory.sinp.msu.ru

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 14, 2015; final manuscript received August 17, 2015; published online October 23, 2015. Assoc. Editor: Gabor Stepan.

J. Comput. Nonlinear Dynam 11(3), 031014 (Oct 23, 2015) (4 pages) Paper No: CND-15-1215; doi: 10.1115/1.4031364 History: Received July 14, 2015; Revised August 17, 2015

Abstract

In this paper, we prove that unviolated simple Leibniz rule and equation for fractional-order derivative of power function cannot hold together for derivatives of orders $α≠1$. To prove this statement, we use an algebraic approach, where special form of fractional-order derivatives is not applied.

<>

References

Samko, S. G. , Kilbas, A. A. , and Marichev, O. I. , 1987, Integrals and Derivatives of Fractional Order and Applications, Nauka i Tehnika, Minsk, Belarus.
Samko, S. G. , Kilbas, A. A. , and Marichev, O. I. , 1993, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York.
Kilbas, A. A. , Srivastava, H. M. , and Trujillo, J. J. , 2003, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands.
Valerio, D. , Trujillo, J. J. , Rivero, M. , Tenreiro Machado, J. A. , and Baleanu, D. , 2013, “ Fractional Calculus: A Survey of Useful Formulas,” Eur. Phys. J., 222(8), pp. 1827–1846.
Tenreiro Machado, J. , Kiryakova, V. , and Mainardi, F. , 2011, “ Recent History of Fractional Calculus,” Commun. Nonlinear Sci. Numer. Simul., 16(3), pp. 1140–1153.
Tenreiro Machado, J. A. , Galhano, A. M. S. F. , and Trujillo, J. J. , 2014, “ On Development of Fractional Calculus During the Last Fifty Years,” Scientometrics, 98(1), pp. 577–582.
Tarasov, V. E. , 2013, “ No Violation of the Leibniz Rule. No Fractional Derivative,” Commun. Nonlinear Sci. Numer. Simul., 18(11), pp. 2945–2948.
Liouville, J. , 1832, “ Memoire sur le Calcul des Differentielles a Indices Quelconques,” J. Ec. R. Polytech., 13, pp. 71–162.
Osler, T. J. , 1970, “ Leibniz Rule for Fractional Derivatives Generalized and an Application to Infinite Series,” SIAM J. Appl. Math., 18(3), pp. 658–674.
Osler, T. J. , 1971, “ Fractional Derivatives and Leibniz Rule,” Am. Math. Mon., 78(6), pp. 645–649.
Osler, T. J. , 1972, “ A Further Extension of the Leibniz Rule to Fractional Derivatives and Its Relation to Parseval's Formula,” SIAM J. Math. Anal., 3(1), pp. 1–16.
Osler, T. J. , 1973, “ A Correction to Leibniz Rule for Fractional Derivatives,” SIAM J. Math. Anal., 4(3), pp. 456–459.
Sabatier, J. , Agrawal, O. P. , and Tenreiro Machado, J. A. , eds., 2007, Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands.
Mainardi, F. , 2010, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, Singapore.
Tarasov, V. E. , 2011, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York.
Gambo, Y. Y. , Jarad, F. , Baleanu, D. , and Abdeljawad, T. , 2014, “ On Caputo Modification of the Hadamard Fractional Derivatives,” Adv. Differ. Equations, 10, pp. 1–12.
Podlubny, I. , 1998, Fractional Differential Equations, Academic Press, San Diego, CA.
Diethelm, K. , 2010, The Analysis of Fractional Differential Equations, Springer, Berlin.
Tarasov, V. E. , 2015, “ 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), pp. 1071–1072.
Jumarie, G. , 2013, “ The Leibniz Rule for Fractional Derivatives Holds With Non-Differentiable Functions,” Math. Stat., 1(2), pp. 50–52.
Weberszpil, J. , 2014, “ Validity of the Fractional Leibniz Rule on a Coarse-Grained Medium Yields a Modified Fractional Chain Rule,” e-print arXiv:1405.4581.
Wang, X. , 2014, “ On the Leibniz Rule and Fractional Derivative for Differentiable and Non-Differentiable Functions,” e-print viXra:1404.0072.
Jumarie, G. , 2006, “ Modified Riemann–Liouville Derivative and Fractional Taylor Series of Non-Differentiable Functions Further Results,” Math. Comput. Appl., 51(9–10), pp. 1367–1376.
Jumarie, G. , 2007, “ Lagrangian Mechanics of Fractional Order, Hamilton–Jacobi Fractional PDE and Taylor's Series of Nondifferentiable Functions,” Chaos, Solitons Fractals, 32(3), pp. 969–987.
Jumarie, G. , 2007, “ The Minkowski's Space–Time is Consistent With Differential Geometry of Fractional Order,” Phys. Lett. A., 363(1–2), pp. 5–11.
Jumarie, G. , 2009, “ Table of Some Basic Fractional Calculus Formulae Derived From a Modified Riemann–Liouville Derivative for Nondifferentiable Functions,” Appl. Math. Lett., 22(3), pp. 378–385.
Jumarie, G. , 2009, “ From Lagrangian Mechanics Fractal in Space to Space Fractal Schrodinger's Equation Via Fractional Taylor's Series,” Chaos, Solitons Fractals, 41(4), pp. 1590–1604.
Jumarie, G. , 2009, “ Probability Calculus of Fractional Order and Fractional Taylor's Series Application to Fokker–Planck Equation and Information of Non-Random Functions,” Chaos, Solitons Fractals, 40(3), pp. 1428–1448.
Jumarie, G. , 2009, “ Oscillation of Non-Linear Systems Close to Equilibrium Position in the Presence of Coarse-Graining in Time and Space,” Nonlinear Anal., 14(2), pp. 177–197.
Jumarie, G. , 2010, “ An Approach Via Fractional Analysis to Non-Linearity Induced by Coarse-Graining in Space,” Nonlinear Anal., 11(1), pp. 535–546.
Jumarie, G. , 2013, “ On the Derivative Chain-Rules in Fractional Calculus Via Fractional Difference and Their Application to Systems Modelling,” Cent. Eur. J. Phys., 11(6), pp. 617–633.
Godinho, C. F. L. , Weberszpil, J. , and Helayel-Neto, J. A. , 2012, “ Extending the D'Alembert Solution to Space–Time Modified Riemann–Liouville Fractional Wave Equations,” Chaos, Solitons Fractals, 45(6), pp. 765–771.
Weberszpil, J. , and Helayel-Neto, J. A. , 2014, “ Anomalous g-factors for Charged Leptons in a Fractional Coarse-Grained Approach,” Adv. High Energy Phys., 2014, p. 572180.
Almeida, R. , and Torres, D. F. M. , 2011, “ Fractional Variational Calculus for Nondifferentiable Functions,” Comput. Math. Appl., 61(10), pp. 3097–3104.
Gomez S., C. A , 2014, “ A Note on the Exact Solution for the Fractional Burgers Equation,” Int. J. Pure Appl. Math., 93(2), pp. 229–232.
Zheng, B. , and Wen, C. , 2013, “ Exact Solutions for Fractional Partial Differential Equations by a New Fractional Sub-Equation Method,” Adv. Differ. Equations, 199, pp. 1–12.
Kolwankar, K. M. , and Gangal, A. D. , 1996, “ Fractional Differentiability of Nowhere Differentiable Functions and Dimensions,” Chaos, 6(4), pp. 505–513. [PubMed]
Kolwankar, K. M. , and Gangal, A. D. , 1997, “ Holder Exponents of Irregular Signals and Local Fractional Derivatives,” Pramana, 48(1), pp. 49–68.
Kolwankar, K. M. , 2013, “ Local Fractional Calculus: A Review,” e-print arXiv:1307.0739.
Ben Adda, F. , and Cresson, J. , 2001, “ About Non-Differentiable Functions,” J. Math. Anal. Appl., 263(2), pp. 721–737.
Liu, C.-S. , 2015, “ Counterexamples on Jumarie's Two Basic Fractional Calculus Formulae,” Commun. Nonlinear Sci. Numer. Simul., 22(1–3), pp. 92–94.
Tarasov, V. E. , 2016, “ On Chain Rule for Fractional Derivatives,” Commun. Nonlinear Sci. Numer. Simul., 30(1–3), pp. 1–4.
Ortigueira, M. D. , and Tenreiro Machado, J. A. , “ What is a Fractional Derivative?,” J. Comput. Phys., 293, pp. 4–13.
Tarasov, V. E. , 2015, “ Local Fractional Derivatives of Differentiable Functions are Integer-Order Derivatives or Zero,” Int. J. Appl. Comput. Math. (in press).
Tarasov, V. E. , 2008, Quantum Mechanics of Non-Hamiltonian and Dissipative Systems, Elsevier Science, New York.
Ammer, C. , 1997, “ Throw Out the Baby With the Bath Water,” The American Heritage Dictionary of Idioms, Houghton Mifflin Harcourt, Boston.
Tarasov, V. E. , 2015, “ Comments on ‘Riemann–Christoffel Tensor in Differential Geometry of Fractional Order Application to Fractal Space–Time,’ [Fractals 21 (2013) 1350004],” Fractals, 23(2), p. 1575001.

Figures

Fig. 1

Plot of the function Z(x) (Eq. (24)) for the range x=α∈[0;1.5]

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections