Investigation of more solitary waves solutions of the stochastics Benjamin-Bona-Mahony equation under beta operator

Abdelkader Moumen; Khaled A. Aldwoah; Muntasir Suhail; Alwaleed Kamel; Hicham Saber; Manel Hleili; Sayed Saifullah; Abdelkader Moumen; Khaled A. Aldwoah; Muntasir Suhail; Alwaleed Kamel; Hicham Saber; Manel Hleili; Sayed Saifullah

doi:10.3934/math.20241331

AIMS Mathematics

2024, Volume 9, Issue 10: 27403-27417. doi: 10.3934/math.20241331

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Investigation of more solitary waves solutions of the stochastics Benjamin-Bona-Mahony equation under beta operator

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 55473, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
3.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
5.
Department of Mathematics and Physics, University of Campania "Luigi Vanvitelli", Caserta 81100, Italy

Received: 27 June 2024 Revised: 28 August 2024 Accepted: 09 September 2024 Published: 23 September 2024
MSC : 35C05, 35C07, 35C08

This study explores the stochastic Benjamin-Bona-Mahony (BBM) equation with a beta derivative (BD), thereby incorporating multiplicative noise in the Itô sense. We derive various analytical soliton solutions for these equations utilizing two distinct expansion methods: the $ \frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}} $-expansion and the modified $ \frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}} $-expansion techniques, both within the framework of beta derivatives. A fractional multistep transformation is employed to convert the equations into nonlinear forms with respect to an independent variable. After performing an algebraic manipulation, the solutions are trigonometric and hyperbolic trigonometric functions. Our analysis demonstrates that the wave behavior is influenced by the fractional-order derivative in the proposed equations, thus providing deeper insights into the wave composition as the fractional order either increases or decreases. Additionally, we explore the effect of white noise on the propagation of the waves solutions. This study underscores the computational robustness and adaptability of the proposed approach to investigate various phenomena in the physical sciences and engineering.
Citation: Abdelkader Moumen, Khaled A. Aldwoah, Muntasir Suhail, Alwaleed Kamel, Hicham Saber, Manel Hleili, Sayed Saifullah. Investigation of more solitary waves solutions of the stochastics Benjamin-Bona-Mahony equation under beta operator[J]. AIMS Mathematics, 2024, 9(10): 27403-27417. doi: 10.3934/math.20241331

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Abstract

This study explores the stochastic Benjamin-Bona-Mahony (BBM) equation with a beta derivative (BD), thereby incorporating multiplicative noise in the Itô sense. We derive various analytical soliton solutions for these equations utilizing two distinct expansion methods: the $ \frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}} $-expansion and the modified $ \frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}} $-expansion techniques, both within the framework of beta derivatives. A fractional multistep transformation is employed to convert the equations into nonlinear forms with respect to an independent variable. After performing an algebraic manipulation, the solutions are trigonometric and hyperbolic trigonometric functions. Our analysis demonstrates that the wave behavior is influenced by the fractional-order derivative in the proposed equations, thus providing deeper insights into the wave composition as the fractional order either increases or decreases. Additionally, we explore the effect of white noise on the propagation of the waves solutions. This study underscores the computational robustness and adaptability of the proposed approach to investigate various phenomena in the physical sciences and engineering.

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