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/TSCI160111018ASuche 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/i2018-12021-3Suche 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.Suche 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.Suche 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.Suche 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.Suche 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.Suche in Google Scholar
[8] R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific, 2nd Ed., 2014.10.1142/8934Suche in Google Scholar
[9] U.N. Katugampola, A new approach to generalized fractional derivatives. Bull. of Mathematical Analysis and Applications6 (2014), 1–15.Suche 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.2014.01.002Suche 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), 505–513.10.1063/1.166197Suche 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.Suche 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.Suche in Google Scholar
[14] R. Magin, Fractional Calculus in Bioengineering. Begell House, Connecticut, 2006.Suche 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.2010.08.003Suche 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/p614Suche 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.Suche 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.Suche 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.Suche in Google Scholar
[20] M.D. Ortigueira, Fractional Calculus for Scientists and Engineers. Lect. Notes in Electr. Engin., Springer, Berlin-Heidelberg, 2011.10.1007/978-94-007-0747-4Suche 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.Suche in Google Scholar
[22] M.D. Ortigueira and J.T. Machado, Which derivative? Fractal and Fractional1, No 3 (2017), 1–13; 10.3390/fractalfract1010003.Suche 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.Suche 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.Suche in Google Scholar
[25] M. Roberts, Signals and Systems: Analysis Using Transform Methods and Matlab. McGraw-Hill, 2 Ed., 2003.Suche in Google Scholar
[26] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach, Yverdon, 1993.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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/s40314-019-0883-8Suche in Google Scholar
[32] D. Valério and J.S. da Costa, An Introduction to Fractional Control. Control Engineering. IET, Stevenage, 2012.10.1049/PBCE091ESuche in Google Scholar
© 2019 Diogenes Co., Sofia
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 22–2–2019)
- Discussion Paper
- The flaw in the conformable calculus: It is conformable because it is not fractional
- The failure of certain fractional calculus operators in two physical models
- Research Paper
- Inverse problems for a class of degenerate evolution equations with Riemann – Liouville derivative
- On Riesz derivative
- A note on fractional powers of strongly positive operators and their applications
- Semi-fractional diffusion equations
- Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations
- Well-posedness of fractional degenerate differential equations in Banach spaces
- Structure factors for generalized grey Browinian motion
- Linear stationary fractional differential equations
- Perfect control for right-invertible Grünwald-Letnikov plants – an innovative approach to practical implementation
- On fractional differential inclusions with Nonlocal boundary conditions
- On solutions of linear fractional differential equations and systems thereof
- Existence of mild solution of a class of nonlocal fractional order differential equation with not instantaneous impulses
- A CAD-based algorithm for solving stable parameter region of fractional-order systems with structured perturbations
- On representation and interpretation of Fractional calculus and fractional order systems
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 22–2–2019)
- Discussion Paper
- The flaw in the conformable calculus: It is conformable because it is not fractional
- The failure of certain fractional calculus operators in two physical models
- Research Paper
- Inverse problems for a class of degenerate evolution equations with Riemann – Liouville derivative
- On Riesz derivative
- A note on fractional powers of strongly positive operators and their applications
- Semi-fractional diffusion equations
- Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations
- Well-posedness of fractional degenerate differential equations in Banach spaces
- Structure factors for generalized grey Browinian motion
- Linear stationary fractional differential equations
- Perfect control for right-invertible Grünwald-Letnikov plants – an innovative approach to practical implementation
- On fractional differential inclusions with Nonlocal boundary conditions
- On solutions of linear fractional differential equations and systems thereof
- Existence of mild solution of a class of nonlocal fractional order differential equation with not instantaneous impulses
- A CAD-based algorithm for solving stable parameter region of fractional-order systems with structured perturbations
- On representation and interpretation of Fractional calculus and fractional order systems