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We take into account the stochastic Boiti–Leon–Manna–Pempinelli equation (SBLMPE), which is perturbed by a multiplicative Brownian motion. By applying He's semi-inverse method and the Riccati equation mapping method, we can acquire the solutions of the SBLMPE. Since the Boiti–Leon–Manna–Pempinelli equation is utilized to explain incompressible liquid in fluid mechanics, the acquired solutions may be applied to explain a lot of fascinating physical phenomena. To address how Brownian motion effects the exact solutions of the SBLMPE, we present some 2D and 3D diagrams. [ABSTRACT FROM AUTHOR]

The stochastic shallow water wave equation (SSWWE) in the sense of the beta-derivative is considered in this study. The solutions of the SSWWE are obtained using the F-expansion technique with the Riccati equation and He's semi-inverse method. Since the shallow water equation has many uses in ocean engineering, including river irrigation flows, tidal waves, tsunami prediction, and weather simulations, the solutions discovered can be utilized to represent a wide variety of exciting physical events. We create many 2D and 3D graphs to demonstrate how the beta-derivative and Brownian motion affect the analytical solutions of the SSWWE. [ABSTRACT FROM AUTHOR]

In this article, the stochastic Davey–Stewartson equations (SDSEs) forced by multiplicative noise are addressed. We use the mapping method to find new rational, elliptic, hyperbolic and trigonometric functions. In addition, we generalize some previously obtained results. Due to the significance of the Davey–Stewartson equations in plasma physics, nonlinear optics, hydrodynamics and other fields, the discovered solutions are useful in explaining a number of intriguing physical phenomena. By using MATLAB tools to simulate our results and display some of 3D graphs, we show how the multiplicative Brownian motion impacts the analytical solutions of the SDSEs. Finally, we demonstrate the effect of multiplicative Brownian motion on the stability and the symmetry of the achieved solutions of the SDSEs. [ABSTRACT FROM AUTHOR]

Fractional–stochastic Drinfel'd–Sokolov–Wilson equations (FSDSWEs) forced by multiplicative Brownian motion are assumed. This equation is employed in mathematical physics, plasma physics, surface physics, applied sciences, and population dynamics. The (G ′ / G) -expansion method is utilized to find rational, hyperbolic, and trigonometric stochastic solutions for FSDSWEs. Because of the priority of FSDSWEs, the derived solutions are more useful and effective in understanding various important physical phenomena. Furthermore, we used the MATLAB package to create 3D graphs for specific solutions in order to investigate the effect of fractional-order and Brownian motions on the solutions of FSDSWEs. [ABSTRACT FROM AUTHOR]

We address here the space-fractional stochastic Hirota–Maccari system (SFSHMs) derived by the multiplicative Brownian motion in the Stratonovich sense. To acquire innovative elliptic, trigonometric and rational stochastic fractional solutions, we employ the Jacobi elliptic functions method. The attained solutions are useful in describing certain fascinating physical phenomena due to the significance of the Hirota–Maccari system in optical fibers. We use MATLAB programm to draw our figures and exhibit several 3D graphs in order to demonstrate how the multiplicative Brownian motion and fractional derivative affect the exact solutions of the SFSHMs. We prove that the solutions of SFSHMs are stabilized by the multiplicative Brownian motion around zero. [ABSTRACT FROM AUTHOR]

A class of stochastic fractional diffusion equations with polynomials is considered in this article. This equation is used in numerous applications, such as ecology, bioengineering, biology, and mechanical and chemical engineering. As a result, it is critical to obtain exact solutions to this equation. To obtain these solutions, the tanh-coth method is utilized. Furthermore, we clarify the impact of noise on solution stabilization by simulating our solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony (BBM) equations. We present a generalised version of the modified tanh–coth method. Using the generalised, modified tanh–coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. With the help of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Eventually, by application example, we show how the stochastic solutions can be given as white noise functional solutions. [ABSTRACT FROM AUTHOR]