Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping

Preeti; Inderdeep Singh

doi:doi:10.11648/j.ajam.20251306.14

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Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping

Preeti^*

, Inderdeep Singh

Published in American Journal of Applied Mathematics (Volume 13, Issue 6)

Received: 29 October 2025 Accepted: 19 November 2025 Published: 17 December 2025

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Abstract

Oscillatory problems are a class of mathematical and physical phenomena in which the solutions display periodic or quasi-periodic variations with respect to time or space. Oscillatory systems subject to external forcing are central in many physical and engineering contexts, and the presence or absence of damping critically influences their behavior. In this comparative study, we develop and evaluate wavelet-based numerical schemes for forced oscillatory differential equations, considering both damped and undamped regimes. Specifically, we employ second-kind Chebyshev wavelets and Haar wavelets, together with their operational integration matrices, to discretize and approximate the solutions of second-order forced oscillators. The wavelet formulations transform the differential problems into systems of algebraic equations, which we then solve under a variety of forcing frequencies and damping parameters. Our numerical experiments demonstrate that Chebyshev wavelets, due to their higher smoothness and spectral accuracy, are particularly effective in capturing the transient decay and subtle features of damped oscillations. In contrast, Haar wavelets provide a computationally efficient and stable approximation in undamped systems, especially in scenarios prone to resonance. We compare these methods across key metrics such as convergence rate, error behavior, and computational cost. The numerical results confirm that Chebyshev wavelets are particularly effective in capturing the fine decay dynamics of damped oscillations, while Haar wavelets deliver fast, stable approximations for undamped systems, especially under resonant forcing. These findings are supported by several computational experiments, which validate both the accuracy and efficiency of the proposed wavelet schemes. These insights offer practical guidance for selecting suitable wavelet approaches tailored to the physical damping conditions of forced oscillatory problems.

Published in	American Journal of Applied Mathematics (Volume 13, Issue 6)
DOI	10.11648/j.ajam.20251306.14
Page(s)	419-427
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Operational Matrices, Chebyshev Wavelets of the Second Kind, Haar Wavelets, Forced Oscillatory Problems, Damped and Undamped Systems

References

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APA Style

Preeti, Singh, I. (2025). Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping. American Journal of Applied Mathematics, 13(6), 419-427. https://doi.org/10.11648/j.ajam.20251306.14

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ACS Style

Preeti; Singh, I. Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping. Am. J. Appl. Math. 2025, 13(6), 419-427. doi: 10.11648/j.ajam.20251306.14

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AMA Style

Preeti, Singh I. Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping. Am J Appl Math. 2025;13(6):419-427. doi: 10.11648/j.ajam.20251306.14

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@article{10.11648/j.ajam.20251306.14,
  author = {Preeti and Inderdeep Singh},
  title = {Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {6},
  pages = {419-427},
  doi = {10.11648/j.ajam.20251306.14},
  url = {https://doi.org/10.11648/j.ajam.20251306.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251306.14},
  abstract = {Oscillatory problems are a class of mathematical and physical phenomena in which the solutions display periodic or quasi-periodic variations with respect to time or space. Oscillatory systems subject to external forcing are central in many physical and engineering contexts, and the presence or absence of damping critically influences their behavior. In this comparative study, we develop and evaluate wavelet-based numerical schemes for forced oscillatory differential equations, considering both damped and undamped regimes. Specifically, we employ second-kind Chebyshev wavelets and Haar wavelets, together with their operational integration matrices, to discretize and approximate the solutions of second-order forced oscillators. The wavelet formulations transform the differential problems into systems of algebraic equations, which we then solve under a variety of forcing frequencies and damping parameters. Our numerical experiments demonstrate that Chebyshev wavelets, due to their higher smoothness and spectral accuracy, are particularly effective in capturing the transient decay and subtle features of damped oscillations. In contrast, Haar wavelets provide a computationally efficient and stable approximation in undamped systems, especially in scenarios prone to resonance. We compare these methods across key metrics such as convergence rate, error behavior, and computational cost. The numerical results confirm that Chebyshev wavelets are particularly effective in capturing the fine decay dynamics of damped oscillations, while Haar wavelets deliver fast, stable approximations for undamped systems, especially under resonant forcing. These findings are supported by several computational experiments, which validate both the accuracy and efficiency of the proposed wavelet schemes. These insights offer practical guidance for selecting suitable wavelet approaches tailored to the physical damping conditions of forced oscillatory problems.},
 year = {2025}
}

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TY  - JOUR
T1  - Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping
AU  - Preeti
AU  - Inderdeep Singh
Y1  - 2025/12/17
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251306.14
DO  - 10.11648/j.ajam.20251306.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 419
EP  - 427
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251306.14
AB  - Oscillatory problems are a class of mathematical and physical phenomena in which the solutions display periodic or quasi-periodic variations with respect to time or space. Oscillatory systems subject to external forcing are central in many physical and engineering contexts, and the presence or absence of damping critically influences their behavior. In this comparative study, we develop and evaluate wavelet-based numerical schemes for forced oscillatory differential equations, considering both damped and undamped regimes. Specifically, we employ second-kind Chebyshev wavelets and Haar wavelets, together with their operational integration matrices, to discretize and approximate the solutions of second-order forced oscillators. The wavelet formulations transform the differential problems into systems of algebraic equations, which we then solve under a variety of forcing frequencies and damping parameters. Our numerical experiments demonstrate that Chebyshev wavelets, due to their higher smoothness and spectral accuracy, are particularly effective in capturing the transient decay and subtle features of damped oscillations. In contrast, Haar wavelets provide a computationally efficient and stable approximation in undamped systems, especially in scenarios prone to resonance. We compare these methods across key metrics such as convergence rate, error behavior, and computational cost. The numerical results confirm that Chebyshev wavelets are particularly effective in capturing the fine decay dynamics of damped oscillations, while Haar wavelets deliver fast, stable approximations for undamped systems, especially under resonant forcing. These findings are supported by several computational experiments, which validate both the accuracy and efficiency of the proposed wavelet schemes. These insights offer practical guidance for selecting suitable wavelet approaches tailored to the physical damping conditions of forced oscillatory problems.
VL  - 13
IS  - 6
ER  -

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Author Information

Preeti

Department of Physical Sciences, Sant Baba Bhag Singh University, Jalandhar, India

Contact Email

http://orcid.org/0000-0003-0168-4187
Inderdeep Singh

Department of Physical Sciences, Sant Baba Bhag Singh University, Jalandhar, India

Contact Email

http://orcid.org/0000-0003-3280-3732

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APA Style

Preeti, Singh, I. (2025). Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping. American Journal of Applied Mathematics, 13(6), 419-427. https://doi.org/10.11648/j.ajam.20251306.14

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ACS Style

Preeti; Singh, I. Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping. Am. J. Appl. Math. 2025, 13(6), 419-427. doi: 10.11648/j.ajam.20251306.14

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AMA Style

Preeti, Singh I. Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping. Am J Appl Math. 2025;13(6):419-427. doi: 10.11648/j.ajam.20251306.14

Copy | Download

@article{10.11648/j.ajam.20251306.14,
  author = {Preeti and Inderdeep Singh},
  title = {Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {6},
  pages = {419-427},
  doi = {10.11648/j.ajam.20251306.14},
  url = {https://doi.org/10.11648/j.ajam.20251306.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251306.14},
  abstract = {Oscillatory problems are a class of mathematical and physical phenomena in which the solutions display periodic or quasi-periodic variations with respect to time or space. Oscillatory systems subject to external forcing are central in many physical and engineering contexts, and the presence or absence of damping critically influences their behavior. In this comparative study, we develop and evaluate wavelet-based numerical schemes for forced oscillatory differential equations, considering both damped and undamped regimes. Specifically, we employ second-kind Chebyshev wavelets and Haar wavelets, together with their operational integration matrices, to discretize and approximate the solutions of second-order forced oscillators. The wavelet formulations transform the differential problems into systems of algebraic equations, which we then solve under a variety of forcing frequencies and damping parameters. Our numerical experiments demonstrate that Chebyshev wavelets, due to their higher smoothness and spectral accuracy, are particularly effective in capturing the transient decay and subtle features of damped oscillations. In contrast, Haar wavelets provide a computationally efficient and stable approximation in undamped systems, especially in scenarios prone to resonance. We compare these methods across key metrics such as convergence rate, error behavior, and computational cost. The numerical results confirm that Chebyshev wavelets are particularly effective in capturing the fine decay dynamics of damped oscillations, while Haar wavelets deliver fast, stable approximations for undamped systems, especially under resonant forcing. These findings are supported by several computational experiments, which validate both the accuracy and efficiency of the proposed wavelet schemes. These insights offer practical guidance for selecting suitable wavelet approaches tailored to the physical damping conditions of forced oscillatory problems.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping
AU  - Preeti
AU  - Inderdeep Singh
Y1  - 2025/12/17
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251306.14
DO  - 10.11648/j.ajam.20251306.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 419
EP  - 427
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251306.14
AB  - Oscillatory problems are a class of mathematical and physical phenomena in which the solutions display periodic or quasi-periodic variations with respect to time or space. Oscillatory systems subject to external forcing are central in many physical and engineering contexts, and the presence or absence of damping critically influences their behavior. In this comparative study, we develop and evaluate wavelet-based numerical schemes for forced oscillatory differential equations, considering both damped and undamped regimes. Specifically, we employ second-kind Chebyshev wavelets and Haar wavelets, together with their operational integration matrices, to discretize and approximate the solutions of second-order forced oscillators. The wavelet formulations transform the differential problems into systems of algebraic equations, which we then solve under a variety of forcing frequencies and damping parameters. Our numerical experiments demonstrate that Chebyshev wavelets, due to their higher smoothness and spectral accuracy, are particularly effective in capturing the transient decay and subtle features of damped oscillations. In contrast, Haar wavelets provide a computationally efficient and stable approximation in undamped systems, especially in scenarios prone to resonance. We compare these methods across key metrics such as convergence rate, error behavior, and computational cost. The numerical results confirm that Chebyshev wavelets are particularly effective in capturing the fine decay dynamics of damped oscillations, while Haar wavelets deliver fast, stable approximations for undamped systems, especially under resonant forcing. These findings are supported by several computational experiments, which validate both the accuracy and efficiency of the proposed wavelet schemes. These insights offer practical guidance for selecting suitable wavelet approaches tailored to the physical damping conditions of forced oscillatory problems.
VL  - 13
IS  - 6
ER  -

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