Numerical exploration of pollutant transport using stochastic fractional diffusion and Karhunen-Loève expansion

This study explores a fractional time-space stochastic diffusion equation for modeling pollutant concentration, incorporating Caputo fractional derivatives and fractional Laplacians to capture anomalous diffusion. Stochastic noise, modeled via Brownian motion and Brownian bridges, is simulated using the Karhunen-Loève expansion. The equation's formulation, along with initial and boundary conditions, is presented. Analytical and numerical methods are discussed, emphasizing a hybrid framework combining Fast Fourier Transform, L1-algorithm, and Karhunen-Loève expansion techniques. Numerical examples with sinc and Gaussian initial conditions highlight the superior accuracy and efficiency of the KLE approach over traditional Euler methods, revealing the significant influence of fractional parameters on pollutant dispersion dynamics and their potential for environmental modeling applications.