This article applied the Riccati–Bernoulli (RB) sub-ODE method in order to get new exact solutions for the long–short-wave interaction (LS) equations. Namely, we obtain deterministic and random solutions, since we consider the proposed method in deterministic and random cases. The RB sub-ODE technique gives the travelling wave solutions in forms of hyperbolic, trigonometric and rational functions. It is shown that the proposed method gives a robust mathematical tool for solving nonlinear wave equations in applied science. Furthermore, some bi-random variables and some random distributions are used in random case corresponding to the LS system. The stability for the obtained solutions in random case is considered. In addition, there is a display of several numerical simulations, which helps to understand the physical phenomena of these soliton wave solutions.

In this work, the conformable fractional derivative with the unified solver method are used in the suitable cases to extract new solutions for space-time stochastic fractional nonlinear Schrödinger equation (NLSE) with spatiotemporal dispersion. Namely, some new stochastic solutions with physical parameters for this equation are constructed via exponential distribution. Exponential distribution is employed to clarify the dispersion random effect. The expectation (mean) of stochastic solutions are depicted to exhibit the effect of random parameters on the solutions of space-time stochastic fractional NLSE with spatiotemporal dispersion. These results are highly applicable to develop new theories of plasma physics, industrial studies, biomedical problems, condense matter physics and optical fibers. Simulations are performed by using Maple software. According to the presented results, the proposed technique can be applied for other fractional models arising in natural sciences.

In the present work, the exp(−φ(ξ))-expansion method is applied for solving the deterministic and stochastic Phi-4 equation. Namely, we introduce hyperbolic, trigonometric, and rational function solutions. The computational study shows that the offered method is pretentious, robust, and influential in applications of interesting analysis, observations of particle physics, plasma physics, quantum field theory, and fluid dynamics. The control on the randomness input (the coefficients are random variables) is studied in order to obtain stability stochastic process solution with beta distribution. In this work, we will deal with stability moment method and then we apply the mean square calculus for the stability concept.

In this paper, relations between moments of dual generalized order statistics from a exponentiated generalized class of distributions, given by Cardeiro et al. (2013) are studied. Some particular cases of dual generalized order statistics and examples based on it are discussed. The characterization of given distribution based on moment properties is also presented.