Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators

• Autorzy: Alharthi Nadiyah Hussain , Atangana Abdon , Alkahtani Badr S.

Identyfikatory

DOI

10.1515/dema-2022-0230

• Języki publikacji: EN

Abstrakty

EN

Nonlocal operators with different kernels were used here to obtain more general harmonic oscillator models. Power law, exponential decay, and the generalized Mittag-Leffler kernels with Delta-Dirac property have been utilized in this process. The aim of this study was to introduce into the damped harmonic oscillator model nonlocalities associated with these mentioned kernels and see the effect of each one of them when computing the Bode diagram obtained from the Laplace and the Sumudu transform. For each case, we applied both the Laplace and the Sumudu transform to obtain a solution in a complex space. For each case, we obtained the Bode diagram and the phase diagram for different values of fractional orders. We presented a detailed analysis of uniqueness and an exact solution and used numerical approximation to obtain a numerical solution.

Słowa kluczowe

EN

damped harmonic oscillator; nonlocal operator; Bode diagram; existence and uniqueness of the solutions; numerical methods

PL

oscylator harmoniczny tłumiony; operator nielokalny; diagram Bodego; istnienie i jednoznaczność rozwiązań; metody numeryczne

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2023
• Tom: Vol. 56, nr 1
• Strony: art. no. 20220230
• Opis fizyczny: Bibliogr. 18 poz., wykr.

Twórcy

autor

Alharthi Nadiyah Hussain

• Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University College of Science, P.O. Box 90950, 11623, Riyadh, Saudi Arabia

autor

Atangana Abdon

• Institute for Groundwater Studies, Faculty of Natural and Agricultural Science, University of the Free State, 9300 Bloemfontein, South Africa

autor

Alkahtani Badr S.

• Department of Mathematics, College of Science, King Saud University, P.O. Box 1142, Riyadh 11989, Saudi Arabia

Bibliografia

• [1] W. B. Case, The pumping of a swing from the standing position, Am. J. Phys. 64 (1996), no. 3, 215–220.
• [2] P. Roura and J. A. Gonzalez, Towards a more realistic description of swing pumping due to the exchange of angular momentum, Eur. J. Phys. 31 (2010), no. 5, 1195–1207.
• [3] G. R. Fowles and G. L. Cassiday, Analytic Mechanics (5th edn.), Saunders College Publishing, Fort Worth, 1986.
• [4] S. I. Hayek, Mechanical vibration and damping, Encycl. Appl. Phys. WILEY-VCH Verlag GmbH & Co KgaA, (15 Apr 2003). DOI: 10.1002/3527600434.eap231.
• [5] P. Tipler, Physics for Scientists and Engineers: Vol. 1 (4th edn.). W. H. Freeman, 1998.
• [6] J. Liouville, Mémoire sur le calcul des différentielles à indices quelconques, J. de. l'École Polytechnique, Paris. 13 (1832), 71–162.
• [7] J. Liouville, Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions, J. de. l'École Polytechnique, Paris. 13 (1832), 1–69.
• [8] M. Caputo, Linear model of dissipation whose Q is almost frequency independent. II, Geophys. J. Int. 13 (1967), no. 5, 529–539.
• [9] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2015), no. 2, 73–85.
• [10] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci. 20 (2016), no. 2, 763–769.
• [11] D. Benson, S. Wheatcraft, and M. Meerschaert. Application of a fractional advection-dispersion equation, Water Resour. Res. 36 (2000), no. 6, 1403–1412.
• [12] Herrmann R, ed. Fractional calculus. Fractional Calculus: An Introduction for Physicists (2nd edn). New Jersey: World Scientific Publishing Co, 54, 2014.
• [13] R. K. R. Yarlagadda, Analog and Digital Signals and Systems. Springer Science & Business Media, 243, 2010.
• [14] M. Van Valkenburg, University of Illinois at Urbana-Champaign, In memoriam: Hendrik W. Bode (1905-1982), IEEE Trans. Autom. Control., AC-29, 3, March 1984, 193–194.
• [15] T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits. (2nd edn). Cambridge UK: Cambridge University Press, p. §14.6 pp. 451–453, 2004.
• [16] W. S. Levine, The control handbook: the electrical engineering handbook series. (2nd edn). Boca Raton FL: CRC Press/IEEE Press. p. §10.1 p. 163, 1996.
• [17] S. Bhatter, K. Jangid, and S. D. Purohit, Fractionalized mathematical models for drug diffusion, Chaos, Solitons Fractals. 165 (2022), 112810.
• [18] M. A. Khan, S. Ullah, and S. Kumar, A robust study on 2019-nCOV outbreaks through non-singular derivative. Eur. Phys. J. Plus. 136, 168.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).