<p>In this study, we focus on the numerical solutions of the Stochastic Gross-Pitaevskii Equation (SGPE), a critical tool for modelling Bose-Einstein Condensates (BEC) in low-temperature regimes. Using a finite difference scheme, we investigate the dynamics of ultra-cold atomic systems under various external potentials and stochastic influences. The potentials explored include harmonic, periodic, Gaussian, magnetic, dipole, and asymmetric (step-like) scalar potentials, as well as systems with attractive and repulsive interactions. Stability and consistency of the numerical approach are demonstrated through Von-Neumann analysis and mean square consistency tests. Simulations incorporating Ornstein-Uhlenbeck noise provide detailed insights into condensate behaviour under these diverse potentials, highlighting the intricate interplay of stochastic effects and interparticle interactions. Our findings offer a comprehensive numerical framework and results that contribute to advancing both theoretical and experimental physics. The trends we quantify (peak reduction and broadening under noise) align qualitatively with observations in trapped-BEC experiments, underscoring the practical applicability of our numerical framework.</p>