Research Article Impact of Multiplicative Noise on the Exact Solutions of the Fractional-Stochastic Boussinesq-Burger System Wael W. Mohammed , 1,2 Farah M. Al-Askar, 3 and M. El-Morshedy 4,5 1 Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt 3 Department of Mathematical Science, Collage of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia 4 Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia 5 Department of Statistics and Computer Science, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Correspondence should be addressed to Wael W. Mohammed; wael.mohammed@mans.edu.eg Received 24 May 2022; Revised 26 August 2022; Accepted 7 September 2022; Published 22 September 2022 Academic Editor: Arzu Akbulut Copyright © 2022 Wael W. Mohammed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we consider the fractional-stochastic Boussinesq-Burger system (FSBBS) generated by the multiplicative Brownian motion. The Jacobi elliptic function techniques are used to create creative elliptic, hyperbolic, and rational fractional-stochastic solutions for FSBBS. Furthermore, we draw 2D and 3D graphs by using the MATLAB Package for some obtained solutions of FSBBS to discuss the influence of the Brownian motion on these solutions. Finally, we indicate that the Brownian motion stabilizes the solutions of FSBBS around zero. 1. Introduction Nonlinear partial differential equations (NLPDEs) have grown in popularity in the area of nonlinear science, owing to their large variety of uses in economics [1], engineering [2], civil engineering [3], soil mechanics [4], physics [5], quan- tum mechanics [6], statistical mechanics [7], solid-state phys- ics [8], population ecology [9], etc. Solitons are among the most common in the setting of NLPDE solutions, and they are essential for understanding nonlinear physical phenom- ena. Solitons are utilized to understand the properties of non- linear media in various areas including quantum electronics, plasma physics, nonlinear optics, and fluid dynamics [10–13]. Recently, the searching of exact soliton solutions to NLPDEs has become an enthralling research topic in engi- neering and applied sciences. Many techniques have been used to determine exact solutions for NLPDE including tanh-sech [14, 15], Darboux transformation [16], sine- cosine [17, 18], exp ð−ϕðςÞÞ-expansion [19], ðG ′ /GÞ-expan- sion [20–22], Lie symmetry analysis method [23], improved F-expansion method [24, 25], Hirota’s function [26], the Jacobi elliptic function [27, 28], and perturbation [29, 30]. The fractional differential equation is extensively used in fluid mechanics, solid state physics, optical fibers, neural physics, quantum field theory, mathematical biology, plasma physics, and other areas [31–34]. Researchers recommend fractional-order derivative over ordinary order derivative because integer-order derivative is essentially a local opera- tor, but fractional-order derivative is so much more. Also, they explain physical phenomena such as quantum mechan- ics, diffusion, gravity, heat, elasticity, fluid dynamics, electro- dynamics, electrostatics, and sound. Recently, the exact solutions with conformable derivative have been obtained in many papers for instance [35–40]. On the other hand, a wide variety of complex nonlinear physical phenomena can be represented using stochastic par- tial differential equations (SPDEs). These kind of equations can be found in many fields, such as physics and finance. Hindawi Journal of Mathematics Volume 2022, Article ID 9288157, 10 pages https://doi.org/10.1155/2022/9288157