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

This work strives to study the impact of both the multiplicative Wiener process and spatial fractional derivatives on the solutions of (3+1) stochastic fractional partial differential equation describing the fluids with gas bubbles. Based on the complete discriminate polynomial system, we introduce a new extension for the auxiliary function method that is more effective for constructing analytical solutions for stochastic fractional partial differential equations and all special cases resulting from it such as deterministic partial differential equations with integer or fractional order derivatives. An algorithm summarizing our method is inserted to avoid confusion about its applicability. The proposed method has multiple advantages such as it furnishes only real solutions and isolates the bounded and unbounded solutions. This method is successfully applied to (3+1) stochastic fractional partial differential equation describing the fluids with gas bubbles and resulting in some new solutions. Some of these solutions are clarified graphically as well as the individual and combined influences of fractional order and noise on some of the obtained solutions. These effects are some changes in the amplitude and the width of the solutions and the smoothness of the surface represented by these solutions.