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RESEARCH ARTICLE

Dynamic behavior of a time-dependent mass harmonic oscillator: Analytical, semi-analytical, and numerical investigation

Shahd Iqtit1† ,  Taqwa Al-Khader2† ,  Olivia Florea3† ,  Jihad Asad4†*
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1 Faculty of Graduate Study, Palestine Technical University–Kadoorie, Tulkarm, Palestin
2 Department of Applied Mathematics, Faculty of Applied Science, Palestine Technical University–Kadoorie, Tulkarm, Palestine
3 Department of Mathematics and Computer Science, Transilvania University of Brasov, Brașov, Romania
4 Department of Physics, Faculty of Applied Science, Palestine Technical University–Kadoorie, Tulkarm, Palestine
†These authors contributed equally to this work.
IJOCTA, 026210089 https://doi.org/10.36922/IJOCTA026210089
Received: 22 May 2026 | Revised: 20 June 2026 | Accepted: 22 June 2026 | Published online: 13 July 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

The harmonic oscillator with time-dependent mass represents an important model for describing dynamical systems in which the mass varies with time due to environmental interaction or internal processes. In this work, the dynamics of a time-dependent mass harmonic oscillator are investigated using analytical, semi-analytical, and numerical approaches. The governing equation of motion is derived using both Newtonian and Lagrangian formulations, with an exponentially decreasing mass function. An exact analytical solution is obtained by transforming the resulting differential equation into a Bessel-type equation. In addition, approximate solutions are computed using the Differential Transform Method (DTM) and its multistep extension (Ms-DTM). The numerical solution obtained with the classical fourth-order Runge–Kutta (RK4) method serves as a reference for evaluating the accuracy of the semi-analytical techniques. The results show excellent agreement between the analytical Bessel solution and the numerical RK4 solution, confirming the validity of the analytical formulation. The Ms-DTM method also provides accurate approximations for short time intervals and demonstrates rapid convergence. These findings indicate that the combined analytical–numerical framework offers a reliable approach for studying the dynamical behavior of harmonic oscillators with time-dependent mass and may be useful in the analysis of engineering systems where variable mass effects are significant.

Keywords
Time-dependent mass
Harmonic oscillator
Bessel function
Multistep differential transform method
Runge–Kutta method
Funding
None.
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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