Analytical and Numerical Investigation of Suspended Sediment Concentration Profiles in an Ice-Covered Channel Using the Time-Fractional Advection–Diffusion Equation

The diffusion behavior of sediments in an ice-covered channel is different from that in a regular open channel due to the presence of an additional ice cover on the upper surface. Several models based on the conventional advection-diffusion equation, as well as on the space-fractional advection-diffusion equation, have been developed to find the concentration profile in such channels. However, the nonlocality characteristics that arise from the time-memory effects of particles remain unaddressed. Therefore, the present study considers the unsteady one-dimensional time-fractional advection-diffusion equation to investigate the time-dependent memory effects of particles in the flow of an ice-covered channel. The modified Atangana-Baleanu fractional derivative in the Caputo sense, involving the Mittag-Leffler kernel with an integrable singularity at the origin, is used. Both analytical and numerical solutions to the problem are presented: analytical solutions using Laplace and Fourier transform methods and numerical solutions using the Chebyshev collocation method. Additionally, a comparison between the analytical and numerical solutions has been conducted, including an error analysis. Furthermore, the effects of time-memory on the concentration profile have been investigated. It has been observed that, as time progresses, sediment particles exhibit a strong memory effect, leading to an increase in their concentration as they move from upstream zones to the current location. During the initial stage of motion, time-memory effects influence particle movement and mixing, but this influence diminishes as the system approaches a steady state. Sensitivity analysis for different parameters present in the model has been conducted, and the model has been validated using existing experimental data under restricted conditions.