Analytical Approaches To Stochastic-Fractional Drinfel’D–Sokolov–Wilson Equations: Improved Modified Sardar Sub-Equation Methods Integrated With Brownian Motion Analysis

The stochastic fractional Drinfel’d–Sokolov–Wilson equations (SFDSWEs) influenced by multiplicative Brownian motion have a vast application in plasma physics, mathematical physics, applied sciences (including sciences of population and surface), and others. This work employs the improved modified Sardar sub-equation method (ISMSSEM) to analyze soliton solutions for these equations. Applying Atangana’s β-derivative, this research focuses on the stochastic-fractional character of SFDSWEs concerning memory and genetic properties. The presented IMSSEM in this paper, from mathematical and epistemological perspectives, provides a comprehensive model of analysis of complex systems with probability and fractional derivatives. The given approach is illustrated by obtaining explicit soliton solutions to nonlinear stochastic-fractional differential equations, thus showing the efficiency and versatility of the method. This work enhances the existing understanding of the SFDSWEs and extends their potential usage in various scientific disciplines. In addition, the presented research contributes to the spectrum of solution methods for fractional differential equations with stochastic characteristics. The IMSSEM distinguishes itself with how easily it can sort and solve equations containing fractional derivatives and those influenced by stochastic effects. In summary, the work extends the utilization of nonlinear stochastic-fractional differential equations and yields practical knowledge of the soliton characteristics of those equations.