On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling

• Authors: Guy Jumarie
• Languages of publication: EN

Abstracts

EN

It has been pointed out that the derivative chains rules in fractional differential calculus via fractional calculus are not quite satisfactory as far as they can yield different results which depend upon how the formula is applied, that is to say depending upon where is the considered function and where is the function of function. The purpose of the present short note is to display some comments (which might be clarifying to some readers) on the matter. This feature is basically related to the non-commutativity of fractional derivative on the one hand, and furthermore, it is very close to the physical significance of the systems under consideration on the other hand, in such a manner that everything is right so. As an example, it is shown that the trivial first order system may have several fractional modelling depending upon the way by which it is observed. This suggests some rules to construct the fractional models of standard dynamical systems, in as meaningful a model as possible. It might happen that this pitfall comes from the feature that a function which is continuous everywhere, but is nowhere differentiable, exhibits random-like features.

Keywords

EN

fractional calculus; fractional Taylor’s series; fractional derivative; systems modelling; fractional derivative chain rule

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 6
• Pages: 617-633

Dates

published

1 - 6 - 2013

online

9 - 10 - 2013

Contributors

author

Guy Jumarie

• Department of Mathematics, University of Quebec at Montreal, P.O. Box 8888, Downtown Station, Montreal Qc, H3C 3P8, Canada

References

• [1] M. Al-Akaidi, Fractal Speech Processing (Cambridge University Press, 2004) http://dx.doi.org/10.1017/CBO9780511754548[Crossref]
• [2] D. Baleanu, S. Vacaru, Fractional analogous models in mechanics and gravity theory, in Fractional Dynamics and Control (Springer, New York, 2012) 16 http://dx.doi.org/10.1007/978-1-4614-0457-6[Crossref]
• [3] D. Baleanu, S. Vacaru, Fractional exact solutions and solitons in Gravity, in Fractional Dynamics and Control (Springer, New York, 2012) 19 http://dx.doi.org/10.1007/978-1-4614-0457-6[Crossref]
• [4] L.M.C. Campos, IMA J. Appl Math 33, 109 (1984) http://dx.doi.org/10.1093/imamat/33.2.109[Crossref]
• [5] L.M.C. Campos, Fractional calculus of analytic and branched functions, in R.N. Kalia (Ed.) (Recent Advances in Fractional Calculus, Global Publishing Company, 1993)
• [6] M. Caputo, Geophys. J. R. Ast. Soc. 13, 529 (1967) http://dx.doi.org/10.1111/j.1365-246X.1967.tb02303.x[Crossref]
• [7] M.M. Djrbashian, A.B. Nersesian, Fractional derivative and the Cauchy problem for differential equations of fractional order 3 (Izv. Acad. Nauk Armjanskoi SSR, 1968) (in Russian)
• [8] C.F.L. Godinho, J. Weberszpil, J.A. Helayël-Nete, Chaos Solit. Fract., DOI: 10.1016/j.chaos.2012.02.008 [Crossref]
• [9] G. Jumarie, Int. J. Syst. Sc. 24, 113 (1993)
• [10] G. Jumarie, Appl. Math. Lett. 18, 739 (2005) http://dx.doi.org/10.1016/j.aml.2004.05.014[Crossref]
• [11] G. Jumarie, Appl. Math. Lett. 18, 817 (2005) http://dx.doi.org/10.1016/j.aml.2004.09.012[Crossref]
• [12] G. Jumarie, Comput. Math. Appl. 51, 1367 (2006) http://dx.doi.org/10.1016/j.camwa.2006.02.001[Crossref]
• [13] G. Jumarie, Math. Comput. Model. 44, 231 (2006) http://dx.doi.org/10.1016/j.mcm.2005.10.003[Crossref]
• [14] G. Jumarie, Chaos Solit. Fract. 32, 969 (2007) http://dx.doi.org/10.1016/j.chaos.2006.07.053[Crossref]
• [15] G. Jumarie, Acta Math. Sinica, DOI: 10.1007/s10114-012-0507-3 [Crossref]
• [16] G. Jumarie, Inf. Sci., DOI:10.1016/j.ins.2012.06.008 [Crossref]
• [17] K.M. Kolwankar, A.D. Gangal, Pramana J. Phys. 48, 49 (1997) http://dx.doi.org/10.1007/BF02845622[Crossref]
• [18] K.M. Kolwankar, A.D. Gangal, Phys. Rev. Lett. 80, 214 (1998) http://dx.doi.org/10.1103/PhysRevLett.80.214[Crossref]
• [19] A.V. Letnikov, Math. Sb. 3, 1 (1868)
• [20] J. Liouville, J. Ecole Polytechnique 13, 71 (1832)
• [21] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional, Differential Equations (Wiley, New York, 1933)
• [22] K. Nishimoto, Fractional Calculus (Descartes Press Co., Koroyama, 1989)
• [23] L. Nottale, Fractal Space Time in Microphyssics (World Scientific, Singapore, 1993) http://dx.doi.org/10.1142/1579[Crossref]
• [24] K.B. Oldham, J. Spanier, The Fractional Calculus, Theory and Application of Differentiation and Integration to Arbitrary Order (Acadenic Press, New York, 1974)
• [25] T.J. Osler, SIAM. J. Math. Anal. 2, 37 (1971) http://dx.doi.org/10.1137/0502004[Crossref]
• [26] I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)
• [27] B. Ross, Fractional Calculus and its Applications, Lectures Notes in Mathematics 457 (Springer, Berlin, 1974)
• [28] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integral and Derivatives, Theory and Applications (Gordon and Breach Science Publishers, London, 1987)

Identifiers

DOI

10.2478/s11534-013-0256-7