The Riccati-Bernoulli sub-optimal differential equation method for analyzing the fractional Dullin-Gottwald-Holm equation and modeling nonlinear waves in fluid mediums

Humaira Yasmin; Haifa A. Alyousef; Sadia Asad; Imran Khan; R. T. Matoog; S. A. El-Tantawy; Humaira Yasmin; Haifa A. Alyousef; Sadia Asad; Imran Khan; R. T. Matoog; S. A. El-Tantawy

doi:10.3934/math.2024781

AIMS Mathematics

2024, Volume 9, Issue 6: 16146-16167. doi: 10.3934/math.2024781

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The Riccati-Bernoulli sub-optimal differential equation method for analyzing the fractional Dullin-Gottwald-Holm equation and modeling nonlinear waves in fluid mediums

1.
Department of Basic Sciences, General Administration of Preparatory Year, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
3.
Department of Physics, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
4.
Department of Architecture and Interior Design, College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia
5.
Department of Mathematics and Statistic, Bacha Khan University, Charsadda, Pakistan
6.
Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah, Saudi Arabia
7.
Department of Physics, Faculty of Science, Port Said University, Port Said 42521, Egypt
8.
Research Center for Physics (RCP), Department of Physics, Faculty of Science and Arts, Al-Mikhwah, Al-Baha University, Al-Baha 1988, Saudi Arabia

Received: 01 February 2024 Revised: 15 April 2024 Accepted: 17 April 2024 Published: 08 May 2024

The present study investigates the fractional Dullin-Gottwald-Holm equation by using the Riccati-Bernoulli sub-optimal differential equation method with the Bäcklund transformation. By employing a well-established criterion, the present study reveals novel cusp soliton solutions that resemble peakons and offers valuable insights into their dynamic behaviors and mysterious phenomena. The solution family encompasses various analytical solutions, such as peakons, periodic, and kink-wave solutions. Furthermore, the impact of both the time- and space-fractional parameters on all derived solutions' profiles is examined. This investigation's significance lies in its contribution to understanding intricate dynamics inside physical systems, offering valuable insights into various domains like fluid mechanics and nonlinear phenomena across different physical models. The computational technique's straightforward, effective, and concise nature is demonstrated through introduction of some graphical representations in two- and three-dimensional plots generated by adjusting the related parameters. The findings underscore the versatility of this methodology and demonstrate its applicability as a tool to solve more complicated nonlinear problems as well as its ability to explain many mysterious phenomena.
Citation: Humaira Yasmin, Haifa A. Alyousef, Sadia Asad, Imran Khan, R. T. Matoog, S. A. El-Tantawy. The Riccati-Bernoulli sub-optimal differential equation method for analyzing the fractional Dullin-Gottwald-Holm equation and modeling nonlinear waves in fluid mediums[J]. AIMS Mathematics, 2024, 9(6): 16146-16167. doi: 10.3934/math.2024781

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Abstract

The present study investigates the fractional Dullin-Gottwald-Holm equation by using the Riccati-Bernoulli sub-optimal differential equation method with the Bäcklund transformation. By employing a well-established criterion, the present study reveals novel cusp soliton solutions that resemble peakons and offers valuable insights into their dynamic behaviors and mysterious phenomena. The solution family encompasses various analytical solutions, such as peakons, periodic, and kink-wave solutions. Furthermore, the impact of both the time- and space-fractional parameters on all derived solutions' profiles is examined. This investigation's significance lies in its contribution to understanding intricate dynamics inside physical systems, offering valuable insights into various domains like fluid mechanics and nonlinear phenomena across different physical models. The computational technique's straightforward, effective, and concise nature is demonstrated through introduction of some graphical representations in two- and three-dimensional plots generated by adjusting the related parameters. The findings underscore the versatility of this methodology and demonstrate its applicability as a tool to solve more complicated nonlinear problems as well as its ability to explain many mysterious phenomena.

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