Piecewise derivatives versus short memory concept: analysis and application

Abdon ATANGANA; Seda İǦRET ARAZ; Abdon ATANGANA; Seda İǦRET ARAZ

doi:10.3934/mbe.2022399

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 8: 8601-8620. doi: 10.3934/mbe.2022399

Previous Article Next Article

Research article Special Issues

Piecewise derivatives versus short memory concept: analysis and application

Abdon ATANGANA ^{1,2
,
,},
Seda İǦRET ARAZ ^1,3

1.
Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa
2.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
3.
Faculty of Education, Siirt University, Siirt 56100, Turkey

Academic Editor: Carlo Cattani

Received: 16 February 2022 Revised: 10 April 2022 Accepted: 05 May 2022 Published: 14 June 2022

We have provided a detailed analysis to show the fundamental difference between the concept of short memory and piecewise differential and integral operators. While the concept of short memory leads to different long tails in different intervals of time or space as a result of a power law with different fractional orders, the concept of piecewise helps to depict crossover behaviors of different patterns. We presented some examples with different numerical simulations. In some cases piecewise models led to transitional behavior from deterministic to stochastic, this is indeed the reason why this concept was introduced.
- piecewise calculus,
- short memory concept,
- crossover behaviors,
- chaos,
- epidemiology
Citation: Abdon ATANGANA, Seda İǦRET ARAZ. Piecewise derivatives versus short memory concept: analysis and application[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 8601-8620. doi: 10.3934/mbe.2022399

Related Papers:

Abstract

We have provided a detailed analysis to show the fundamental difference between the concept of short memory and piecewise differential and integral operators. While the concept of short memory leads to different long tails in different intervals of time or space as a result of a power law with different fractional orders, the concept of piecewise helps to depict crossover behaviors of different patterns. We presented some examples with different numerical simulations. In some cases piecewise models led to transitional behavior from deterministic to stochastic, this is indeed the reason why this concept was introduced.

References

[1]	M. Caputo, Linear model of dissipation whose Q is almost frequency independent-Ⅱ, Geophys. J. Int., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
[2]	T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7–15.
[3]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[4]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.98/TSCI160111018A doi: 10.98/TSCI160111018A
[5]	Podlubny I., Fractional differential equations, mathematics in science and engineering, Academic Press, 198 (1999).
[6]	Y. Zhou, Y. Zhang, Noether symmetries for fractional generalized Birkhoffian systems in terms of classical and combined Caputo derivatives, Acta Mech., 231 (2020), 3017–3029. https://doi.org/10.1007/s00707-020-02690-y doi: 10.1007/s00707-020-02690-y
[7]	W. Sumelka, B. Łuczaka, T. Gajewskia, G.Z. Voyiadjis, Modelling of AAA in the framework of time-fractional damage hyperelasticity, Int. J. Solids Struct., 206 (2020), 30–42. https://doi.org/10.1016/j.ijsolstr.2020.08.015 doi: 10.1016/j.ijsolstr.2020.08.015
[8]	J. Sabatier, Fractional-order derivatives defined by continuous kernels: Are they really too restrictive?, Fractal Fract., 4 (2020), 40. https://doi.org/10.3390/fractalfract4030040 doi: 10.3390/fractalfract4030040
[9]	G. C. Wu, Z. G. Deng, D. Baleanu, D. Q. Zeng, Fractional impulsive differential equations: Exact solutions, integral equations and short memory case, Fract. Calc. Appl. Anal., 22 (2019). https://doi.org/10.1515/fca-2019-0012 doi: 10.1515/fca-2019-0012
[10]	A. Atangana, S. Igret Araz, New concept in calculus: Piecewise differential and integral operators, Chaos Solit. Fract., 145 (2021). https://doi.org/10.1016/j.chaos.2020.110638 doi: 10.1016/j.chaos.2020.110638
[11]	W. H. Deng, Short memory principle and a predictor-corrector approach for fractional differential equations, J. Comput. Appl. Math., 206 (2007), 174–188. https://doi.org/10.1016/j.cam.2006.06.008 doi: 10.1016/j.cam.2006.06.008
[12]	A. Atangana, S. İğret Araz, Advanced analysis in epidemiological modeling: Detection of wave, MedRixv, (2021). https://doi.org/10.1101/2021.09.02.21263016 doi: 10.1101/2021.09.02.21263016
[13]	B. Ghanbari, D. Kumar, Numerical solution of predator-prey model with Beddington-DeAngelis functional response and fractional derivatives with Mittag-Leffler kernel, Chaos, 29 (2019). https://doi.org/10.1063/1.5094546 doi: 10.1063/1.5094546
[14]	G. Qi, G. Chen, M. A. Van Myk, B. J. Van Myk, Y. Zhang, A four-wing chaotic attractor generated from a new 3-D quadratic chaotic system, Chaos Solit. Fractals., 38 (2008), 705–721. https://doi.org/10.1016/j.chaos.2007.01.029 doi: 10.1016/j.chaos.2007.01.029
[15]	G. Qi, Z. Wang, Y. Guo, Generation of an eight-wing chaotic attractor from Qi 3-D four-wing chaotic system, Int. J. Bifurc. Chaos, 22 (2012). https://doi.org/10.1142/S0218127412502872 doi: 10.1142/S0218127412502872
[16]	T. Mekkoui, A. Atangana New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, Eur. Phys. J. Plus, 132 (2017). https://doi.org/10.1140/epjp/s13360-022-02380-9 doi: 10.1140/epjp/s13360-022-02380-9
[17]	A. Atangana, S. Igret Araz, New numerical scheme with Newton polynomial: Theory, Methods and Applications, Academic Press, (2021). https://doi.org/10.1016/B978-0-12-775850-3.50017-0

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)