Approximate symmetries and conservation laws of forced fractional oscillator

Document Type : Research Paper


1 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16846--13114, Iran

2 Department of Mathematics, Payame Noor University, Tehran 19395--4697, Iran


The approximate equation for the forced fractional oscillator is obtained by approximation of the Riemann- Liouville fractional derivatives. And the approximate symmetries and conservation laws of the forced fractional oscillator are derived when the system is in resonance.


[1] R. Caponetto, G. Dongola, L. Fortuna and I. Petrasi, Fractional order systems.Modeling and control applications, World Scientific Series on Nonlinear Science, Series A, 2010.

[2] R. K. Gazizov and S.Yu. Lukashchuk, Approximations of fractional differential equations and approximate symmetries, IFAC Papers OnLine 50 (2017), 14022–14027.

[3] R. Herrmann, Fractional Calculus. An introduction for Physicists, World Scientific Publishing Co., 2014.

[4] N.H. Ibragimov, V.F. Kovalev, Approximate and renormgroup symmetries, Nonlinear Phys. Sci., Springer, 2009.

[5] N.H. Ibragimov, Nonlinear self-adjointness and conservation laws. J. Phys. A: Math Theor. 44 (2011), no. 43, 1–11.

[6] N.H. Ibragimov, Nonlinear self-adjointness in constructing conservation laws, arXiv:1109.1728 [math-ph] (2011). Arch Alga 7/8.

[7] R.C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51 (1984), no. 2, 299–307.

[8] P. Labedzki, R. Pawlikowski and A. Radowicz, Axial vibrations of bars using fractional viscoelastic material models, Vib. Phys. Syst. 29 (2018), 1–8.

[9] P. Labedzki, R. Pawlikowski and A. Radowiczi, Transverse vibration of a cantilever beam under base excitation using fractional rheological model, AIP Conf. Proc., Kielce University of Technology, 2018, pp. 1–10.

[10] P. Labedzki, R. Pawlikowski and A. Radowicz, On fractional forced oscillator, AIP Conf. Proc., Kielce University of Technology, 2019, pp. 1–9.
[11] Y.J. Lee, Vibrations and Waves, 8.03SC Physics III, MIT Open Course Ware, 2016.

[12] S.Yu. Lukashchuk, Approximate conservation laws for fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 68 (2019), 147–159.

[13] F. Mainardi, Fractional calculus and waves in Linear viscoelasticity, Imperial College Press, 2005.

[14] O. Martin, Nonlinear dynamic analysis of viscoelastic beams using a fractional rheological model, Appl. Math. Model. 43 (2017), 351–359.

[15] M. Nadjafikhah and A. Mokhtary, Approximate symmetry analysis of Gardner equation, arXiv:1212.3604 (2012) [math.AP] 14.

[16] S. Rogosin and F. Mainardi, George William Scott-Blair – the pioneer of fractional calculus in rheology, npreprint arXiv:1404 .3295 (2014) 1–22.

[17] J. Sabatier, O.P Agrawal and J.A Tenreiro Machado, Advances in fractional calculus, Theoretical Developments and Applications in Physics and Engineering, Springer Dordrecht, 2007.

[18] J. Yuan, Y. Zhang, J. Liu, B. Shi, M. Gai and S. Yang, Mechanical energy and equivalent differential equations of motion for single-degree-of-freedom fractional oscillators, J. Sound Vib. 397 (2017), 192–203.
Volume 14, Issue 2
February 2023
Pages 195-205
  • Receive Date: 20 January 2022
  • Revise Date: 01 August 2022
  • Accept Date: 09 November 2022