Application of Newton's polynomial interpolation scheme for variable order fractional derivative with power-law kernel.
Atangana–Toufik scheme
Chaotic system
Rikitake system
Rucklidge system
Variable order fractional derivative
Wang–Sun system
Journal
Scientific reports
ISSN: 2045-2322
Titre abrégé: Sci Rep
Pays: England
ID NLM: 101563288
Informations de publication
Date de publication:
12 Jul 2024
12 Jul 2024
Historique:
received:
04
03
2024
accepted:
02
07
2024
medline:
13
7
2024
pubmed:
13
7
2024
entrez:
12
7
2024
Statut:
epublish
Résumé
This paper, offers a new method for simulating variable-order fractional differential operators with numerous types of fractional derivatives, such as the Caputo derivative, the Caputo-Fabrizio derivative, the Atangana-Baleanu fractal and fractional derivative, and the Atangana-Baleanu Caputo derivative via power-law kernels. Modeling chaotical systems and nonlinear fractional differential equations can be accomplished with the utilization of variable-order differential operators. The computational structures are based on the fractional calculus and Newton's polynomial interpolation. These methods are applied to different variable-order fractional derivatives for Wang-Sun, Rucklidge, and Rikitake systems. We illustrate this novel approach's significance and effectiveness through numerical examples.
Identifiants
pubmed: 38997322
doi: 10.1038/s41598-024-66494-z
pii: 10.1038/s41598-024-66494-z
doi:
Types de publication
Journal Article
Langues
eng
Sous-ensembles de citation
IM
Pagination
16090Informations de copyright
© 2024. The Author(s).
Références
Diethelm, K. & Ford, N. The analysis of fractional differential equations. Lect. Notes Math 2004, 3–12 (2010).
doi: 10.1007/978-3-642-14574-2_1
Kilbas, A. A., Srivastava, H. M. & Trujillo, J. J. Theory and applications of fractional differential equations Vol. 204 (Elsevier, 2006).
doi: 10.1016/S0304-0208(06)80001-0
Lakshmikantham, V. & Vatsala, A. S. Basic theory of fractional differential equations. Nonlinear Anal. Theory, Methods Appl. 69, 2677–2682 (2008).
doi: 10.1016/j.na.2007.08.042
Hilfer, R. Applications of fractional calculus in physics (World scientific, 2000).
doi: 10.1142/3779
Atangana, A. & Araz, S. İ. New numerical scheme with Newton polynomial: theory, methods, and applications (Academic Press, 2021).
Caputo, M. & Fabrizio, M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85 (2015).
Toufik, M. & Atangana, A. New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models. Eur. Phys. J. Plus 132, 1–16 (2017).
doi: 10.1140/epjp/i2017-11717-0
Atangana, A. & Araz, S. İ. New numerical approximation for chua attractor with fractional and fractal-fractional operators. Alex. Eng. J. 59, 3275–3296 (2020).
doi: 10.1016/j.aej.2020.01.004
Almutairi, N. & Saber, S. Application of a time-fractal fractional derivative with a power-law kernel to the burke-shaw system based on newton’s interpolation polynomials. Methods 12, 102510 (2024).
Ahmed, K. I., Adam, H. D., Almutairi, N. & Saber, S. Analytical solutions for a class of variable-order fractional liu system under time-dependent variable coefficients. Results in Phys. 56, 107311 (2024).
doi: 10.1016/j.rinp.2023.107311
Ramirez, L. E. & Coimbra, C. F. A variable order constitutive relation for viscoelasticity. Ann. der Phys. 519, 543–552 (2007).
doi: 10.1002/andp.200751907-803
Sun, H., Chen, W. & Chen, Y. Variable-order fractional differential operators in anomalous diffusion modeling. Phys. Stat. Mech. Appl. 388, 4586–4592 (2009).
doi: 10.1016/j.physa.2009.07.024
Chauhan, A., Gautam, G., Chauhan, S. & Dwivedi, A. A validation on concept of formula for variable order integral and derivatives. Chaos, Solitons Fractals 169, 113297 (2023).
doi: 10.1016/j.chaos.2023.113297
Sun, H., Chang, A., Zhang, Y. & Chen, W. A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications. Fract. Calc. Appl. Anal. 22, 27–59 (2019).
doi: 10.1515/fca-2019-0003
Ma, S. et al. Numerical solutions of a variable-order fractional financial system. J. Appl. Math. 20(1), 417942 (2012).
doi: 10.1155/2012/417942
Naveen, S. & Parthiban, V. Qualitative analysis of variable-order fractional differential equations with constant delay. Math. Methods Appl. Sci. 47, 2981–2992 (2023).
doi: 10.1002/mma.9789
Naik, M. K., Baishya, C., Veeresha, P. & Baleanu, D. Design of a fractional-order atmospheric model via a class of act-like chaotic system and its sliding mode chaos control: Chaos an interdisciplinary. J. Nonlinear Sci. https://doi.org/10.1063/5.0130403 (2023).
doi: 10.1063/5.0130403
Baishya, C., Premakumari, R., Samei, M. E. & Naik, M. K. Chaos control of fractional order nonlinear bloch equation by utilizing sliding mode controller. Chaos, Solitons Fractals 174, 113773 (2023).
doi: 10.1016/j.chaos.2023.113773
Baishya, C., Naik, M. K. & Premakumari, R. Design and implementation of a sliding mode controller and adaptive sliding mode controller for a novel fractional chaotic class of equations. Results Control Optim. 14, 100338 (2024).
doi: 10.1016/j.rico.2023.100338
Naik, M. K., Baishya, C. & Veeresha, P. A chaos control strategy for the fractional 3d lotka-volterra like attractor. Math. Comput. Simul. 211, 1–22 (2023).
doi: 10.1016/j.matcom.2023.04.001
Dubey, V. P., Kumar, D., Alshehri, H. M., Singh, J. & Baleanu, D. Generalized invexity and duality in multiobjective variational problems involving non-singular fractional derivative. Open Phys. 20, 939–962 (2022).
doi: 10.1515/phys-2022-0195
Atangana, A. & Araz, S. İ. New numerical method for ordinary differential equations: newton polynomial. Elsevier https://doi.org/10.1016/j.cam.2019.112622 (2020).
doi: 10.1016/j.cam.2019.112622
Bezzateev, S., Davydov, V. & Ometov, A. On secret sharing with newton’s polynomial for multi-factor authentication. Cryptography 4, 34 (2020).
doi: 10.3390/cryptography4040034
Gandha, G. I. & Santoso, D. A. The newton’s polynomial based-automatic model generation (amg) for sensor calibration to improve the performance of the low-cost ultrasonic range finder (hc-sr04). J. Infotel 12, 115–122 (2020).
doi: 10.20895/infotel.v12i3.486
Heydari, M. H., Atangana, A. & Avazzadeh, Z. Chebyshev polynomials for the numerical solution of fractal-fractional model of nonlinear Ginzburg–Landau equation. Eng. Comput. 37, 1377–1388 (2021).
doi: 10.1007/s00366-019-00889-9
Almutairi, N. & Saber, S. Application of a time-fractal fractional derivative with a power-law kernel to the Burke–Shaw system based on newton’s interpolation polynomials. Methods 12, 102510 (2024).
Toufik, M. & Atangana, A. New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models. Eur. Phys. J. Plus 132, 1–16 (2017).
doi: 10.1140/epjp/i2017-11717-0
Zou, L. et al. A new approach to newton-type polynomial interpolation with parameters. Math. Probl. Eng. 2020, 1–15 (2020).
Babayar-Razlighi, B. Newton–Taylor polynomial solutions of systems of nonlinear differential equations with variable coefficients. Int. J. Nonlinear Anal. Appl. 12, 237–248 (2021).
Dadras, S. & Momeni, H. R. A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Phys. Lett. A 373, 3637–3642 (2009).
doi: 10.1016/j.physleta.2009.07.088
Wang, Z., Sun, Y., van Wyk, B. J., Qi, G. & van Wyk, M. A. A 3-d four-wing attractor and its analysis. Braz. J. Phys. 39, 547–553 (2009).
doi: 10.1590/S0103-97332009000500007
Alkahtani, B. S. T. A new numerical scheme based on newton polynomial with application to fractional nonlinear differential equations. Alex. Eng. J. 59, 1893–1907 (2020).
doi: 10.1016/j.aej.2019.11.008
Atangana, A. & Qureshi, S. Modeling attractors of chaotic dynamical systems with fractal-fractional operators. Chaos, Solitons Fractals 123, 320–337 (2019).
doi: 10.1016/j.chaos.2019.04.020
Dubey, V. P., Singh, J., Dubey, S. & Kumar, D. Some integral transform results for Hilfer–Prabhakar fractional derivative and analysis of free-electron laser equation. Iran. J. Sci. 47, 1333–1342 (2023).
doi: 10.1007/s40995-023-01493-9
Dubey, V. P., Singh, J., Dubey, S. & Kumar, D. Analysis of cauchy problems and diffusion equations associated with the Hilfer–Prabhakar fractional derivative via kharrat-toma transform. Fractal Fract. 7, 413 (2023).
doi: 10.3390/fractalfract7050413
Singh, J., Ghanbari, B., Dubey, V. P., Kumar, D. & Nisar, K. S. Fractional dynamics and computational analysis of food chain model with disease in intermediate predator. AIMS Math. 9, 17089–17121 (2024).
doi: 10.3934/math.2024830
Singh, J., Dubey, V. P., Kumar, D., Dubey, S. & Baleanu, D. Fractal-view analysis of local fractional Fokker–Planck equation occurring in modelling of particle’s Brownian motion. Opt. Quantum Electron. 56, 1109 (2024).
doi: 10.1007/s11082-024-06842-5
Singh, J., Jassim, H. K., Kumar, D. & Dubey, V. P. Fractal dynamics and computational analysis of local fractional Poisson equations arising in electrostatics. Commun. Theor. Phys. 75, 125002 (2023).
doi: 10.1088/1572-9494/ad01ad
Dubey, V. P., Singh, J., Alshehri, A. M., Dubey, S. & Kumar, D. Analysis and fractal dynamics of local fractional partial differential equations occurring in physical sciences. J. Comput. Nonlinear Dyn. 18, 031001 (2023).
doi: 10.1115/1.4056360
Ito, K. Chaos in the Rikitake two-disc dynamo system. Earth Planet. Sci. Lett. 51, 451–456 (1980).
doi: 10.1016/0012-821X(80)90224-1
Llibre, J. & Messias, M. Global dynamics of the rikitake system. Phys. D Nonlinear Phenom. 238, 241–252 (2009).
doi: 10.1016/j.physd.2008.10.011
Lăzureanu, C. & Binzar, T. A rikitake type system with quadratic control. Int. J. Bifurc. Chaos 22, 1250274 (2012).
doi: 10.1142/S0218127412502744
Ramanathan, C. et al. A new chaotic attractor from rucklidge system and its application in secured communication using ofdm. In 2017 11th International Conference on Intelligent Systems and Control (ISCO), 241–245 (IEEE, 2017).
Kocamaz, U. E. & Uyaroğlu, Y. Controlling Rucklidge Chaotic system with a single controller using linear feedback and passive control methods. Nonlinear Dyn. 75, 63–72 (2014).
doi: 10.1007/s11071-013-1049-7
Solís-Pérez, J., Gómez-Aguilar, J. & Atangana, A. Novel numerical method for solving variable-order fractional differential equations with power, exponential and mittag-leffler laws. Chaos Solitons Fractals 114, 175–185 (2018).
doi: 10.1016/j.chaos.2018.06.032
Chen, G. & Ueta, T. Yet another chaotic attractor. Int. J. Bifurc. chaos 9, 1465–1466 (1999).
doi: 10.1142/S0218127499001024
Chen, G. Lj. dynamical analysis, control and synchronization of the generalized lorenz systems family (2003).
Rikitake, T. Oscillations of a system of disk dynamos. In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 54, 89–105 (Cambridge University Press, 1958).