Analytical solutions to the fractional stochastic (3 + 1) equation of fluids with gas bubbles using an extended auxiliary function method

Exploring the Exact Solution of the Space-Fractional Stochastic Regularized Long Wave Equation: A Bifurcation Approach

“…In this section, we aim to study some of the solutions obtained previously, especially the ones in ( 21) and (25). We will investigate the distinct effects of fractional order and noise on these solutions, both separately and in combination.…”

“…Using both fractional and stochastic generalizations in one model offers a more flexible tool for modeling systems that exhibit both memory effects and randomness. Research on stochastic fractional differential equations (SFDEs) is a growing area in applied mathematics, e.g., [20][21][22][23][24][25][26][27].…”

“…For f = −0.25 ∈ ( f 1 , 0) = (−0.5759999997, 0), Equation ( 33) has a solution (25) which is given by…”