The Analytical Solutions for the Stochastic-Fractional Broer–Kaup Equations

Abstract

We take into account here the stochastic-fractional Broer–Kaup equations (SFBKEs) perturbed by the multiplicative Wiener process. To get rational, hyperbolic, and elliptic stochastic solutions for SBKEs, we utilize the Jacobi elliptic function method. The derived solutions are significantly more useful and effective in comprehending various important challenging physical phenomena due to the important of SFBKEs in describing the propagation of shallow water waves. Also, we use the MATLAB Package to create 2D and 3D graphs for certain solutions of SFBKEs in order to discuss the impact of fractional order and the Wiener process on the solutions of SFBKEs.

Bifurcation analysis on ion sound and Langmuir solitary waves solutions to the stochastic models with multiplicative noises

“…This study is devoted to highlighting the wave behaviors and the effect of noise on the ion sound wave solutions of model (1), which was studied through He's semi-inverse, the Riccati–Bernoulli sub-ODE and the sine–cosine techniques by Mohammad et al [ 38 ]. They used only auxiliary schemes to find stochastic solutions, but we derived solution following energy orbits from Hamilatonian of the model.…”

“…Among these methods, the bifurcation as well as theory of dynamic structures approach are one most qualitative as this can derive the exact solutions according to the energy orbits of their phase portraits. Besides, stochastic soliton of stochastic nonlinear models are highly investigated by recent young scientists [ [36] , [37] , [38] , [39] ] to reducing arbitrary fluctuations of wave's amplitudes. It is still unexplored the stochastic wave solution following each energy orbits of phase portraits from Hamiltonian of the model.…”

Impact of Fractional Derivative and Brownian Motion on the Solutions of the Radhakrishnan-Kundu-Lakshmanan Equation

“…Obtaining solutions of PDEs is crucial for understanding physical phenomena. Therefore, many effective methods, including exp-function method [1], auxiliary equation [2], Darboux transformation [3], sine-cosine [4], Jacobi elliptic function [5], exp ð−ϕðςÞÞ-expansion [6], sine-Gordon expansion [7], ðG ′ /GÞ-expansion [8][9][10], generalized Kudryashov [11], perturbation [12][13][14], extended trial equation [15,16], Jacobi elliptic function [17,18], Riccati equation [19], tanh-coth [20], homotopy perturbation [21], modified decomposition [22], and F-expansion [23], have been constructed to attain exact solutions of PDEs.…”

Exact Solutions of the Stochastic Conformable Broer–Kaup Equations

“…In this study, we investigated the exact solutions of an equation with a stochastic fractional derivative. In this case, we also search for stochastic conformable Broer-Kaup equations for a better understanding of random wave states [18]:…”