Sediment loss (SLO) is a critical geological phenomenon characterized by a reduction in mass over time due to the combined effects of controlling variables such as rock competency, weathering, and erosion. This study aims to develop an efficient numerical method for solving SLO fractal-fractional differential equations (SLO-FFDEs) based on shifted Legendre polynomials (SLPs) and to investigate the effects of the fractal, fractional, and erosion parameters on the sediment loss dynamics. The SLO-FFDEs are approximated using the basis vectors of SLPs, and a derivative operational matrix (OM) of SLPs is developed within the framework of the new generalized Caputo fractal-fractional derivative (GCFFD). The proposed method is evaluated for different values of the fractal parameter σ, fractional parameter ϱ, and decaying parameter λ associated with erosion. The numerical results demonstrate that the proposed OM-based framework provides an effective and accurate approach for describing the dynamics of sediment loss under different combinations of fractal and fractional parameters. The results also indicate that variations in the fractal and fractional parameters significantly influence the behavior of the SLO model, while the erosion parameter controls the rate of sediment loss. Therefore, the proposed method provides a flexible computational framework for modeling SLO-FFDEs and can be used to investigate the influence of fractal, fractional, and erosion effects in sediment-loss processes. It is recommended that the proposed framework be further extended to more complex sediment-loss models and validated using experimental or field data.

Sediment loss (SLO) is a critical geological phenomenon characterized by a reduction in mass over time due to the combined effects of controlling variables such as rock competency, weathering, and erosion. This study aims to develop an efficient numerical method for solving SLO fractal-fractional differential equations (SLO-FFDEs) based on shifted Legendre polynomials (SLPs) and to investigate the effects of the fractal, fractional, and erosion parameters on the sediment loss dynamics. The SLO-FFDEs are approximated using the basis vectors of SLPs, and a derivative operational matrix (OM) of SLPs is developed within the framework of the new generalized Caputo fractal-fractional derivative (GCFFD). The proposed method is evaluated for different values of the fractal parameter σ, fractional parameter ϱ, and decaying parameter λ associated with erosion. The numerical results demonstrate that the proposed OM-based framework provides an effective and accurate approach for describing the dynamics of sediment loss under different combinations of fractal and fractional parameters. The results also indicate that variations in the fractal and fractional parameters significantly influence the behavior of the SLO model, while the erosion parameter controls the rate of sediment loss. Therefore, the proposed method provides a flexible computational framework for modeling SLO-FFDEs and can be used to investigate the influence of fractal, fractional, and erosion effects in sediment-loss processes. It is recommended that the proposed framework be further extended to more complex sediment-loss models and validated using experimental or field data.