In this research, we are examining the stochastic modified Korteweg-de Vries (SMKdV) equation forced in the Itô sense by multiplicative noise. We use an appropriate transformation to convert the SMKdV equation to another MKdV equation with random variable coefficients (MKdV-RVCs). We use the generalizing Riccati equation mapping and Jacobi elliptic functions methods in order to acquire new trigonometric, hyperbolic, and rational solutions for MKdV-RVCs. After that, we can get the solutions to the SMKdV equation. To our knowledge, this is the first time we have assumed that the solution of the wave equation for the SMKdV equation is stochastic, since all earlier research assumed that it was deterministic. Furthermore, we provide different graphic representations to show the influence of multiplicative noise on the exact solutions of the SMKdV equation.

In this paper, we introduce novel extensions of the reversed Minkowski inequality for various functions defined on time scales. Our approach involves the application of Jensen's and Hölder's inequalities on time scales. Our results encompass the continuous inequalities established by Benaissa as special cases when the time scale $\mathbb{T}$ corresponds to the real numbers (when $\mathbb{T}\mathbb{=}\mathbb{R}$). Additionally, we derive distinct inequalities within the realm of time scale calculus, such as cases $\mathbb{T}\mathbb{=}\mathbb{N}$ and $q^{\mathbb{N}}$ for $q>1$. These findings represent new and significant contributions for the reader.

In this paper, we establish some new dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus by applying the reverse H ölder's inequality, integration by parts, and chain rule on time scales nabla calculus. As special cases of our results (when $\mathbb{T}\mathbb{=}\mathbb{R}$), we get the continuous analouges of inequalities proven by Benaissa and Sarikaya, and when ${\mathbb{T}\mathbb{=}\mathbb{N}}_0$, the results to the best of the authors' knowledge are essentially new.

The fractional-stochastic Fokas-Lenells equation (FSFLE) in the Stratonovich sense is taken into account here. The modified mapping method is used to generate new trigonometric, hyperbolic, elliptic and rational stochastic fractional solutions. Because the Fokas-Lenells equation has many implementations in telecommunication modes, complex system theory, quantum field theory, and quantum mechanics, the obtained solutions can be employed to describe a wide range of exciting physical phenomena. We plot several 2D and 3D diagrams to demonstrate how multiplicative noise and fractional derivatives affect the analytical solutions of the FSFLE. Also, we show how multiplicative noise at zero stabilizes FSFLE solutions.

Here, we consider the stochastic Kakutani–Matsuuchi model (SKMM) perturbed by multiplicative noise in the Itô sense. This model describes the behavior of these waves as they propagate through a stratified fluid medium, such as the Earth's atmosphere or ocean. Internal gravity waves are generated by disturbances in the density or temperature of the fluid and can play a significant role in transporting energy and momentum throughout the system. By applying two different techniques, namely the extended tanh function method and the mapping method, we obtain new periodic soliton, dark soliton, bright soliton, anti-Kink soliton and Kink soliton solutions for SKMM. Because the Kakutani–Matsuuchi model is important in studying internal gravity waves in the atmosphere and oceans, the solutions of the SKMM are beneficial in understanding several fascinating scientific phenomena. Using MATLAB, we exhibit several 2D and 3D graphs that illustrate the impact of the noise on the solutions of SKMM.