Home The failure of certain fractional calculus operators in two physical models
Article
Licensed
Unlicensed Requires Authentication

The failure of certain fractional calculus operators in two physical models

  • Manuel D. Ortigueira EMAIL logo , Valeriy Martynyuk , Mykola Fedula and J. Tenreiro Machado
Published/Copyright: May 11, 2019

Abstract

The ability of the so-called Caputo-Fabrizio (CF) and Atangana-Baleanu (AB) operators to create suitable models for real data is tested with real world data. Two alternative models based on the CF and AB operators are assessed and compared with known models for data sets obtained from electrochemical capacitors and the human body electrical impedance. The results show that the CF and AB descriptions perform poorly when compared with the classical fractional derivatives.

Acknowledgements

The authors thank to colleagues V. De Santis, P. A. Beeckman, D. A. Lampasi, and M. Feliziani [5] for supplying with the experimental results that were the base of our modeling of the human body electrical impedance.

This work was funded by Portuguese National Funds through the FCT – Foundation for Science and Technology under the Project PEst-UID/EEA/00066/2013.

References

[1] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model. Thermal Science20, No 2 (2016), 763–769.10.2298/TSCI160111018ASearch in Google Scholar

[2] A. Atangana and J.F. Gómez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena. The European Physical J. Plus133, No 4 (Apr 2018), Art. # 166, 22 pp.10.1140/epjp/i2018-12021-3Search in Google Scholar

[3] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, No 2 (2015), 73–85.Search in Google Scholar

[4] V. De Santis, P.A. Beeckman, D.A. Lampasi, and M. Feliziani, Assessment of human body impedance for safety requirements against contact currents for frequencies up to 110 MHz. IEEE Trans. on Biomedical Engineering58, No 2 (Feb 2011), 390–396; 10.1109/TBME.2010.2066273.Search in Google Scholar PubMed

[5] V. De Santis, V. Martynyuk, A. Lampasi, M. Fedula, and M. Ortigueira, Fractional-order circuit models of the human body impedance for compliance tests against contact currents. AEU - International J. of Electronics and Communications78 (2017), 238–244; 10.1016/j.aeue.2017.04.035.Search in Google Scholar

[6] A. Flexner, The usefulness of useless knowledge. Harper’s Magazine, Issue No 179 (1939), 544–552; at https://library.ias.edu/files/UsefulnessHarpers.pdf.Search in Google Scholar

[7] A. Giusti, A comment on some new definitions of fractional derivative. Nonlinear Dynamics93, No 3 (Aug 2018), 1757–1763; 10.1007/s11071-018-4289-8.Search in Google Scholar

[8] R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific, 2nd Ed., 2014.10.1142/8934Search in Google Scholar

[9] U.N. Katugampola, A new approach to generalized fractional derivatives. Bull. of Mathematical Analysis and Applications6 (2014), 1–15.Search in Google Scholar

[10] R. Khalil, M.A. Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative. J. of Computational and Applied Mathematics264 (2014), 65–70.10.1016/j.cam.2014.01.002Search in Google Scholar

[11] K.M. Kolwankar and A.D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions. Chaos: An Interdisciplinary J. of Nonlinear Science6, No 4 (1996), 505–513.10.1063/1.166197Search in Google Scholar PubMed

[12] J.A.T. Machado, And I say to myself: “What a fractional world !”. Fract. Calc. Appl. Anal. 14, No 4 (2011), 635–654; 10.2478/s13540-011-0037-1; https://www.degruyter.com/view/j/fca.2011.14.issue-4/issue-files/fca.2011.14.issue-4.xml.Search in Google Scholar

[13] J.A.T. Machado and V. Kiryakova, The chronicles of fractional calculus. Fract. Calc. Appl. Anal. 20, No 2 (2017), 307–336; 10.1515/fca-2017-0017; https://www.degruyter.com/view/j/fca.2017.20.issue-2/issue-files/fca.2017.20.issue-2.xml.Search in Google Scholar

[14] R. Magin, Fractional Calculus in Bioengineering. Begell House, Connecticut, 2006.Search in Google Scholar

[15] R. Magin, M.D. Ortigueira, I. Podlubny, and J. Trujillo, On the fractional signals and systems. Signal Processing91, No 3 (2011), 350–3711.10.1016/j.sigpro.2010.08.003Search in Google Scholar

[16] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London, 2010.10.1142/p614Search in Google Scholar

[17] V. Martynyuk and M. Ortigueira, Fractional model of an electrochemical capacitor. Signal Processing107 (2015), 355–360; 10.1016/j.sigpro.2014.02.021.Search in Google Scholar

[18] V. Martynyuk, M. Ortigueira, M. Fedula, and O. Savenko, Methodology of electrochemical capacitor quality control with fractional order model. AEU - International J. of Electronics and Communications91 (2018), 118–124; 10.1016/j.aeue.2018.05.005.Search in Google Scholar

[19] E.C. Oliveira and J.A.T. Machado, A review of definitions for fractional derivatives and integrals. Mathematical Problems in Engineering2014, No 3 (2014), Art. # 238459, 6 pp; 10.1155/2014/238459.Search in Google Scholar

[20] M.D. Ortigueira, Fractional Calculus for Scientists and Engineers. Lect. Notes in Electr. Engin., Springer, Berlin-Heidelberg, 2011.10.1007/978-94-007-0747-4Search in Google Scholar

[21] M.D. Ortigueira and J.T. Machado, What is a fractional derivative? J. of Computational Physics293 (2015), 4–13; 10.1016/j.jcp.2014.07.019.Search in Google Scholar

[22] M.D. Ortigueira and J.T. Machado, Which derivative? Fractal and Fractional1, No 3 (2017), 1–13; 10.3390/fractalfract1010003.Search in Google Scholar

[23] M.D. Ortigueira and J.T. Machado, A critical analysis of the Caputo-Fabrizio operator. Commun. in Nonl. Sci. and Numer. Simul. 59 (2018), 608–611; 10.1016/j.cnsns.2017.12.001.Search in Google Scholar

[24] M.D. Ortigueira and J.T. Machado, Fractional derivatives: The perspective of system theory. Mathematics7, No 2 (2019), Art. # 150, 14 pp.; 10.3390/math7020150.Search in Google Scholar

[25] M. Roberts, Signals and Systems: Analysis Using Transform Methods and Matlab. McGraw-Hill, 2 Ed., 2003.Search in Google Scholar

[26] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach, Yverdon, 1993.Search in Google Scholar

[27] J.V.C. Sousa and E.C. de Oliveira, Mittag-Leffler functions and the truncated 𝓥-fractional derivative. Mediterranean J. of Math. 14, No 6 (Nov 2017), Art. # 244, 26 pp.; 10.1007/s00009-017-1046-z.Search in Google Scholar

[28] J.V.C. Sousa and E.C. de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties. International J. of Analysis and Applications 16, No 1 (2018), 83–96.Search in Google Scholar

[29] M. Stynes, Fractional-order derivatives defined by continuous kernels are too restrictive. Applied Mathematics Letters85 (2018), 22–26; 10.1016/j.aml.2018.05.013.Search in Google Scholar

[30] V.E. Tarasov, No nonlocality. No fractional derivative. Commun. in Nonl. Sci. and Numer. Simul. 62 (2018), 157–163; 10.1016/j.cnsns.2018.02.019.Search in Google Scholar

[31] V.E. Tarasov, Caputo-Fabrizio operator in terms of integer derivatives: Memory or distributed lag? Computational and Applied Mathematics, 2019.10.1007/s40314-019-0883-8Search in Google Scholar

[32] D. Valério and J.S. da Costa, An Introduction to Fractional Control. Control Engineering. IET, Stevenage, 2012.10.1049/PBCE091ESearch in Google Scholar

Received: 2019-01-18
Published Online: 2019-05-11
Published in Print: 2019-04-24

© 2019 Diogenes Co., Sofia

Downloaded on 16.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2019-0017/html
Scroll to top button