Soliton solutions of the time-fractional Poisson-Nernst-Planck system with stochastic analysis and their application

This study investigated the influence of Brownian motion and noise effects on the dynamics of the stochastic Poisson-Nernst-Planck system with M-truncated fractional derivative. To explore exact analytical representations of the soliton solutions, the study employed the modified extended direct algebraic method. The method successfully produces closed-form exact soliton solutions that capture the stochastic behavior of the system in the presence of random perturbations. The addition of the M-truncated fractional derivative provides a more flexible structure to describe anomalous transport, offering an advanced mathematical representation of electro-diffusion processes. The obtained results highlight the combined role of noise, fractional dynamics, and stochastic fluctuations in shaping the system's evolution, thereby deepening the theoretical understanding of nonlinear stochastic transport models and opening potential avenues for applications in complex biological and physical systems. Moreover, the study presented graphical demonstrations that illustrate the effect of noise and the fractional order of derivation in 3D, 2D, and contour surfaces.