An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation

Myong Hyok Sin; Ryu Song Jo

doi:doi:10.11648/j.ml.20261201.11

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An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation

Myong Hyok Sin^*

, Ryu Song Jo

Published in Mathematics Letters (Volume 12, Issue 1)

Received: 19 October 2025 Accepted: 4 November 2025 Published: 7 January 2026

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Abstract

In this paper, we proposed a new approximate solution method of the Bagley- Torvik fractional order differential equation with damping term. The solution of this equation is of great practical interest and has been studied extensively. In general, the solution of the Bagley-Torvik equation is known to be computationally expensive and the process is complicated. However, the new proposed method significantly reduced the computational effort while ensuring the accuracy of the solution in a novel way. Since 1.5 order derivative of absolutely continuous a function on some interval in the Caputo’s sense is equal to the temperature gradient at the boundary of the one-dimensional heat conduction problem in the semi-infinite interval with 1 order derivative of the function as boundary conditions, we transformed the given fractional differential equation into a general differential equation. Then, according to the characteristics of the obtained equation, we used the variables separation and Fourier series approximation. So the proposed method transforms the Bagley-Torvik fractional order differential equation into an integer order differential equation. Prior to the main content, we have given and proved two definitions and two theorems as preliminaries. And the convergence analysis of this method is discussed. And then we solved two different example problems for the cases with and without damping by using various methods including proposed method and verified that the computational effort is significantly small and the accuracy is guaranteed through comparison of the results with other methods. So the effectiveness of the proposed method is analyzed.

Published in	Mathematics Letters (Volume 12, Issue 1)
DOI	10.11648/j.ml.20261201.11
Page(s)	1-11
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), 2026. Published by Science Publishing Group

Keywords

Bagley-Torvik Equation, Fractional Order Differential Equation, Fourier Series Approximation

References

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[9]	S. Saha Ray, R. K. Bera, Analytical solution of the Bagley Torvik equation by Adomian decomposition method, Applied Mathematics and Computation 168 (2005) 398-410. https://doi.org/10.1016/j.amc.2004.09.006
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[15]	Mustafa Gülsu, Yalçın Öztürk, Ayşe Anapali, Numerical solution of the fractional Bagley-Torvik equation arising in fluid mechanics, International Journal of Computer Mathematics. http://dx.doi.org/10.1080/00207160.2015.1099633
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APA Style

Sin, M. H., Jo, R. S. (2026). An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation. Mathematics Letters, 12(1), 1-11. https://doi.org/10.11648/j.ml.20261201.11

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ACS Style

Sin, M. H.; Jo, R. S. An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation. Math. Lett. 2026, 12(1), 1-11. doi: 10.11648/j.ml.20261201.11

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AMA Style

Sin MH, Jo RS. An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation. Math Lett. 2026;12(1):1-11. doi: 10.11648/j.ml.20261201.11

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@article{10.11648/j.ml.20261201.11,
  author = {Myong Hyok Sin and Ryu Song Jo},
  title = {An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation},
  journal = {Mathematics Letters},
  volume = {12},
  number = {1},
  pages = {1-11},
  doi = {10.11648/j.ml.20261201.11},
  url = {https://doi.org/10.11648/j.ml.20261201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20261201.11},
  abstract = {In this paper, we proposed a new approximate solution method of the Bagley- Torvik fractional order differential equation with damping term. The solution of this equation is of great practical interest and has been studied extensively. In general, the solution of the Bagley-Torvik equation is known to be computationally expensive and the process is complicated. However, the new proposed method significantly reduced the computational effort while ensuring the accuracy of the solution in a novel way. Since 1.5 order derivative of absolutely continuous a function on some interval in the Caputo’s sense is equal to the temperature gradient at the boundary of the one-dimensional heat conduction problem in the semi-infinite interval with 1 order derivative of the function as boundary conditions, we transformed the given fractional differential equation into a general differential equation. Then, according to the characteristics of the obtained equation, we used the variables separation and Fourier series approximation. So the proposed method transforms the Bagley-Torvik fractional order differential equation into an integer order differential equation. Prior to the main content, we have given and proved two definitions and two theorems as preliminaries. And the convergence analysis of this method is discussed. And then we solved two different example problems for the cases with and without damping by using various methods including proposed method and verified that the computational effort is significantly small and the accuracy is guaranteed through comparison of the results with other methods. So the effectiveness of the proposed method is analyzed.},
 year = {2026}
}

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TY - JOUR
T1 - An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation
AU - Myong Hyok Sin
AU - Ryu Song Jo
Y1 - 2026/01/07
PY - 2026
N1 - https://doi.org/10.11648/j.ml.20261201.11
DO - 10.11648/j.ml.20261201.11
T2 - Mathematics Letters
JF - Mathematics Letters
JO - Mathematics Letters
SP - 1
EP - 11
PB - Science Publishing Group
SN - 2575-5056
UR - https://doi.org/10.11648/j.ml.20261201.11
AB - In this paper, we proposed a new approximate solution method of the Bagley- Torvik fractional order differential equation with damping term. The solution of this equation is of great practical interest and has been studied extensively. In general, the solution of the Bagley-Torvik equation is known to be computationally expensive and the process is complicated. However, the new proposed method significantly reduced the computational effort while ensuring the accuracy of the solution in a novel way. Since 1.5 order derivative of absolutely continuous a function on some interval in the Caputo’s sense is equal to the temperature gradient at the boundary of the one-dimensional heat conduction problem in the semi-infinite interval with 1 order derivative of the function as boundary conditions, we transformed the given fractional differential equation into a general differential equation. Then, according to the characteristics of the obtained equation, we used the variables separation and Fourier series approximation. So the proposed method transforms the Bagley-Torvik fractional order differential equation into an integer order differential equation. Prior to the main content, we have given and proved two definitions and two theorems as preliminaries. And the convergence analysis of this method is discussed. And then we solved two different example problems for the cases with and without damping by using various methods including proposed method and verified that the computational effort is significantly small and the accuracy is guaranteed through comparison of the results with other methods. So the effectiveness of the proposed method is analyzed.
VL - 12
IS - 1
ER -

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Myong Hyok Sin

Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea

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http://orcid.org/0009-0009-0994-5469
Ryu Song Jo

Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea

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APA Style

Sin, M. H., Jo, R. S. (2026). An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation. Mathematics Letters, 12(1), 1-11. https://doi.org/10.11648/j.ml.20261201.11

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ACS Style

Sin, M. H.; Jo, R. S. An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation. Math. Lett. 2026, 12(1), 1-11. doi: 10.11648/j.ml.20261201.11

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AMA Style

Sin MH, Jo RS. An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation. Math Lett. 2026;12(1):1-11. doi: 10.11648/j.ml.20261201.11

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@article{10.11648/j.ml.20261201.11,
  author = {Myong Hyok Sin and Ryu Song Jo},
  title = {An Efficient Numerical Solution Method of the Bagley-Torvik Equation with Damping Term by Equivalent Transform and Series Approximation},
  journal = {Mathematics Letters},
  volume = {12},
  number = {1},
  pages = {1-11},
  doi = {10.11648/j.ml.20261201.11},
  url = {https://doi.org/10.11648/j.ml.20261201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20261201.11},
  abstract = {In this paper, we proposed a new approximate solution method of the Bagley- Torvik fractional order differential equation with damping term. The solution of this equation is of great practical interest and has been studied extensively. In general, the solution of the Bagley-Torvik equation is known to be computationally expensive and the process is complicated. However, the new proposed method significantly reduced the computational effort while ensuring the accuracy of the solution in a novel way. Since 1.5 order derivative of absolutely continuous a function on some interval in the Caputo’s sense is equal to the temperature gradient at the boundary of the one-dimensional heat conduction problem in the semi-infinite interval with 1 order derivative of the function as boundary conditions, we transformed the given fractional differential equation into a general differential equation. Then, according to the characteristics of the obtained equation, we used the variables separation and Fourier series approximation. So the proposed method transforms the Bagley-Torvik fractional order differential equation into an integer order differential equation. Prior to the main content, we have given and proved two definitions and two theorems as preliminaries. And the convergence analysis of this method is discussed. And then we solved two different example problems for the cases with and without damping by using various methods including proposed method and verified that the computational effort is significantly small and the accuracy is guaranteed through comparison of the results with other methods. So the effectiveness of the proposed method is analyzed.},
 year = {2026}
}

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