Relaxation and Oscillation Model Using Caputo Fractional Differential Equations


Wildy Ardan(1), Siti Fatimah(2), Kartika Yulianti(3*)

(1) Indonesia University of Education, Indonesia
(2) Indonesia University of Education, Indonesia
(3) Indonesia University of Education, Indonesia
(*) Corresponding Author

Abstract


The phenomenon of relaxation and oscillation is a common event that is often encountered. Both of these properties can occur in viscoelastic materials even though they do not occur simultaneously. Because the characteristics of viscoelastic materials are difficult to describe using classical-order differential equations, in this study, fractional-order differential equations were used to model each of the relaxation and oscillation phenomena in viscoelastic materials with the help of Laplace transform as a solution method . The solution obtained characterizes the phenomenon of memory effect as well as viscoelastic materials in general. In addition to this phenomenon, several other variables were also found to be the influence of the related material motion dynamics.

Keywords


Caputo fractional derivative, fractional derivative, oscillation, relaxation, viscoelastic

Full Text:

PDF

References


J. C. Maxwell, On the Dynamical Theory of Gases. 2003. doi: 10.1142/9781848161337_0014.

Soedojo and Peter, Basic Physics. Yogyakarta: Andi Publisher Yogyakarta, 1999.

M. A. Meyers and K. K. Chawla, Mechanical behavior of materials. Cambridge university press, 2008.

R. M. Christensen, Theory of Viscoelasticity. London: Academic Press Publisher, 1971.

W. Chen, X.-D. Zhang, and D. Korošak, “Investigation on fractional and fractal derivative relaxation-oscillation models,” Int. J. Nonlinear Sci. Numer. Simul., vol. 11, no. 1, pp. 3–10, 2010.

I. Podlubny, Fractional Differential Equations. San Diego: ”, Academic Press Publishers, 1999.

S. S. Ray, A. Atangana, S. C. Noutchie, M. Kurulay, N. Bildik, and A. Kilicman, “Fractional calculus and its applications in applied mathematics and other sciences,” Mathematical Problems in Engineering, vol. 2014. Hindawi, 2014.

A. Atangana, Fractional operators with constant and variable order with application to geo- hydrology. Academic Press, 2017.

A. Alshabanat, M. Jleli, S. Kumar, and B. Samet, “Generalization of Caputo-Fabrizio fractional derivative and applications to electrical circuits,” Front. Phys., vol. 8, p. 64, 2020.

Hijrah and D. A. Rahmatul, The Caputo fractional derivative Equation and Its Application to the Semi-Infinite Cooling Process by Radiation. Surabaya: Sepuluh Nopember Institute of Technology, 2016.

M. Caputo, “Linear models of dissipation whose Q is almost frequency independent—II,” Geophys. J. Int., vol. 13, no. 5, pp. 529–539, 1967.

G. V Brown and C. M. North, “The impact damped harmonic oscillator in free decay,” 1987.




DOI: https://doi.org/10.15575/kubik.v7i2.17929

Refbacks

  • There are currently no refbacks.


Copyright (c) 2023 Kartika Yulianti, Siti Fatimah, Wildy Ardan

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.


Journal KUBIK: Jurnal Publikasi Ilmiah Matematika has indexed by:

SINTA DOAJ Dimensions Google Scholar Garuda Moraref DOI Crossref

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.