Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators

Yinlin Ye; Hongtao Fan; Yajing Li

doi:doi:10.11648/j.acm.20251405.11

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Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators

Yinlin Ye, Hongtao Fan

, Yajing Li^*

Published in Applied and Computational Mathematics (Volume 14, Issue 5)

Received: 29 August 2025 Accepted: 8 September 2025 Published: 25 September 2025

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Abstract

The space-time fractional Benjamin-Ono equation (STFBOE) is of fundamental importance in ocean science, particularly for modeling wave propagation in deep water. This study investigates the STFBOE employing three distinct fractional operators: the conformable derivative, the beta derivative, and the M-truncated derivative. By applying a fractional traveling wave transformation, the original nonlinear fractional partial differential equation is reduced to an ordinary differential equation. We then utilize three analytical techniques-the fractional functional variable method, the modified Kudryashov method, and the improved F-expansion method to derive novel exact traveling wave solutions. To the best of our knowledge, the exact solutions for the fractional Benjamin-Ono equation in the form considered here have been scarcely studied. A key contribution of this work is the introduction and tailored application of these methods to systematically construct solutions under each fractional derivative definition. Accordingly, we establish specific traveling wave variables corresponding to the conformable, beta, and M-truncated derivatives. A significant advantage of the proposed framework is its flexibility and effectiveness, enabling a straightforward derivation of solutions for both time-fractional and space-fractional versions of the Benjamin-Ono equation. Finally, we present a comparative graphical analysis of the obtained solutions using two- and three-dimensional plots, illustrating the spatio-temporal dynamics under selected parameter values. All the derived solutions are entirely new and extend the current understanding of nonlinear wave phenomena described by the STFBOE.

Published in	Applied and Computational Mathematics (Volume 14, Issue 5)
DOI	10.11648/j.acm.20251405.11
Page(s)	253-263
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

Space-time Fractional Benjamin-Ono Equation, Beta Derivative, M-truncated derivatives, Fractional Functional Variable Method, Modified Kudryashov Method, Improved F-expansion Method

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APA Style

Ye, Y., Fan, H., Li, Y. (2025). Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators. Applied and Computational Mathematics, 14(5), 253-263. https://doi.org/10.11648/j.acm.20251405.11

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ACS Style

Ye, Y.; Fan, H.; Li, Y. Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators. Appl. Comput. Math. 2025, 14(5), 253-263. doi: 10.11648/j.acm.20251405.11

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AMA Style

Ye Y, Fan H, Li Y. Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators. Appl Comput Math. 2025;14(5):253-263. doi: 10.11648/j.acm.20251405.11

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@article{10.11648/j.acm.20251405.11,
  author = {Yinlin Ye and Hongtao Fan and Yajing Li},
  title = {Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {5},
  pages = {253-263},
  doi = {10.11648/j.acm.20251405.11},
  url = {https://doi.org/10.11648/j.acm.20251405.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251405.11},
  abstract = {The space-time fractional Benjamin-Ono equation (STFBOE) is of fundamental importance in ocean science, particularly for modeling wave propagation in deep water. This study investigates the STFBOE employing three distinct fractional operators: the conformable derivative, the beta derivative, and the M-truncated derivative. By applying a fractional traveling wave transformation, the original nonlinear fractional partial differential equation is reduced to an ordinary differential equation. We then utilize three analytical techniques-the fractional functional variable method, the modified Kudryashov method, and the improved F-expansion method to derive novel exact traveling wave solutions. To the best of our knowledge, the exact solutions for the fractional Benjamin-Ono equation in the form considered here have been scarcely studied. A key contribution of this work is the introduction and tailored application of these methods to systematically construct solutions under each fractional derivative definition. Accordingly, we establish specific traveling wave variables corresponding to the conformable, beta, and M-truncated derivatives. A significant advantage of the proposed framework is its flexibility and effectiveness, enabling a straightforward derivation of solutions for both time-fractional and space-fractional versions of the Benjamin-Ono equation. Finally, we present a comparative graphical analysis of the obtained solutions using two- and three-dimensional plots, illustrating the spatio-temporal dynamics under selected parameter values. All the derived solutions are entirely new and extend the current understanding of nonlinear wave phenomena described by the STFBOE.
},
 year = {2025}
}

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TY  - JOUR
T1  - Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators

AU  - Yinlin Ye
AU  - Hongtao Fan
AU  - Yajing Li
Y1  - 2025/09/25
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251405.11
DO  - 10.11648/j.acm.20251405.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 253
EP  - 263
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251405.11
AB  - The space-time fractional Benjamin-Ono equation (STFBOE) is of fundamental importance in ocean science, particularly for modeling wave propagation in deep water. This study investigates the STFBOE employing three distinct fractional operators: the conformable derivative, the beta derivative, and the M-truncated derivative. By applying a fractional traveling wave transformation, the original nonlinear fractional partial differential equation is reduced to an ordinary differential equation. We then utilize three analytical techniques-the fractional functional variable method, the modified Kudryashov method, and the improved F-expansion method to derive novel exact traveling wave solutions. To the best of our knowledge, the exact solutions for the fractional Benjamin-Ono equation in the form considered here have been scarcely studied. A key contribution of this work is the introduction and tailored application of these methods to systematically construct solutions under each fractional derivative definition. Accordingly, we establish specific traveling wave variables corresponding to the conformable, beta, and M-truncated derivatives. A significant advantage of the proposed framework is its flexibility and effectiveness, enabling a straightforward derivation of solutions for both time-fractional and space-fractional versions of the Benjamin-Ono equation. Finally, we present a comparative graphical analysis of the obtained solutions using two- and three-dimensional plots, illustrating the spatio-temporal dynamics under selected parameter values. All the derived solutions are entirely new and extend the current understanding of nonlinear wave phenomena described by the STFBOE.

VL  - 14
IS  - 5
ER  -

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Yinlin Ye

College of Science, Northwest A&F University, Xianyang, China

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Hongtao Fan

College of Science, Northwest A&F University, Xianyang, China

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http://orcid.org/0000-0003-1079-3094
Yajing Li

College of Science, Northwest A&F University, Xianyang, China

Contact Email

http://orcid.org/0000-0002-8014-2302

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Ye, Y., Fan, H., Li, Y. (2025). Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators. Applied and Computational Mathematics, 14(5), 253-263. https://doi.org/10.11648/j.acm.20251405.11

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ACS Style

Ye, Y.; Fan, H.; Li, Y. Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators. Appl. Comput. Math. 2025, 14(5), 253-263. doi: 10.11648/j.acm.20251405.11

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AMA Style

Ye Y, Fan H, Li Y. Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators. Appl Comput Math. 2025;14(5):253-263. doi: 10.11648/j.acm.20251405.11

Copy | Download

@article{10.11648/j.acm.20251405.11,
  author = {Yinlin Ye and Hongtao Fan and Yajing Li},
  title = {Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {5},
  pages = {253-263},
  doi = {10.11648/j.acm.20251405.11},
  url = {https://doi.org/10.11648/j.acm.20251405.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251405.11},
  abstract = {The space-time fractional Benjamin-Ono equation (STFBOE) is of fundamental importance in ocean science, particularly for modeling wave propagation in deep water. This study investigates the STFBOE employing three distinct fractional operators: the conformable derivative, the beta derivative, and the M-truncated derivative. By applying a fractional traveling wave transformation, the original nonlinear fractional partial differential equation is reduced to an ordinary differential equation. We then utilize three analytical techniques-the fractional functional variable method, the modified Kudryashov method, and the improved F-expansion method to derive novel exact traveling wave solutions. To the best of our knowledge, the exact solutions for the fractional Benjamin-Ono equation in the form considered here have been scarcely studied. A key contribution of this work is the introduction and tailored application of these methods to systematically construct solutions under each fractional derivative definition. Accordingly, we establish specific traveling wave variables corresponding to the conformable, beta, and M-truncated derivatives. A significant advantage of the proposed framework is its flexibility and effectiveness, enabling a straightforward derivation of solutions for both time-fractional and space-fractional versions of the Benjamin-Ono equation. Finally, we present a comparative graphical analysis of the obtained solutions using two- and three-dimensional plots, illustrating the spatio-temporal dynamics under selected parameter values. All the derived solutions are entirely new and extend the current understanding of nonlinear wave phenomena described by the STFBOE.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators

AU  - Yinlin Ye
AU  - Hongtao Fan
AU  - Yajing Li
Y1  - 2025/09/25
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251405.11
DO  - 10.11648/j.acm.20251405.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 253
EP  - 263
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251405.11
AB  - The space-time fractional Benjamin-Ono equation (STFBOE) is of fundamental importance in ocean science, particularly for modeling wave propagation in deep water. This study investigates the STFBOE employing three distinct fractional operators: the conformable derivative, the beta derivative, and the M-truncated derivative. By applying a fractional traveling wave transformation, the original nonlinear fractional partial differential equation is reduced to an ordinary differential equation. We then utilize three analytical techniques-the fractional functional variable method, the modified Kudryashov method, and the improved F-expansion method to derive novel exact traveling wave solutions. To the best of our knowledge, the exact solutions for the fractional Benjamin-Ono equation in the form considered here have been scarcely studied. A key contribution of this work is the introduction and tailored application of these methods to systematically construct solutions under each fractional derivative definition. Accordingly, we establish specific traveling wave variables corresponding to the conformable, beta, and M-truncated derivatives. A significant advantage of the proposed framework is its flexibility and effectiveness, enabling a straightforward derivation of solutions for both time-fractional and space-fractional versions of the Benjamin-Ono equation. Finally, we present a comparative graphical analysis of the obtained solutions using two- and three-dimensional plots, illustrating the spatio-temporal dynamics under selected parameter values. All the derived solutions are entirely new and extend the current understanding of nonlinear wave phenomena described by the STFBOE.

VL  - 14
IS  - 5
ER  -

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