Water pollution is a significant threat for human health, particularly in developed countries. This study advances the mathematical understanding of WP transmission dynamics by developing a fractional–fractal derivative framework with non-singular kernels and the Mittage–Leffler function, which successfully preserves the non-local behavior of pollutants. The fractional–fractal derivatives in sense of the Atangana–Baleanu–Caputo formulation inherently captures the non-local and memory-dependent behavior of pollutant diffusion, addressing limitations of classical differential operators. A novel parameter, $\gamma$, is introduced to represent the recovery rate of water systems through treatment processes, explicitly modeling the bridge between natural purification mechanisms and engineered remediation efforts. Furthermore, this study establishes stability analysis, and the existence and uniqueness of the solution are established through fixed-point theory to ensure the mathematical stability of the system. Moreover, a numerical scheme based on the Newton polynomial is formulated, by obtaining significant simulations of pollution dynamics under various conditions. Graphical results show the effect of important parameters on pollutant evolution, providing useful information about the behavior of the system.

In this paper, we discuss skew circulant matrices involving the product of Pell and Pell-Lucas numbers. The invertibility of the skew circulant matrices is investigated, while the fundamental theorems on the determinants and inverses of such matrices are derived by simple construction matrices. Specifically, the determinant and inverse of \(n\times n\) skew circulant matrices can be expressed by the \((n-1)\)th, \(n\)th, \((n+1)\)th, \((n+2)\)th product of Pell and Pell-Lucas numbers. Some norms and bounds for spread of these matrices are given, respectively. In addition, we generalized these results to skew left circulant matrix involving the product of Pell and Pell-Lucas numbers. Finally, several numerical examples are illustrated to show the effectiveness of our theoretical results.

In this paper, we focused on solving the perturbed four-banded linear system derived from the traffic process associated with a Markovian queueing model. Utilizing the spectral decomposition of circulant and skew circulant matrices, we computed the product of Toeplitz inversion and a vector, leading to a decomposition algorithm for perturbed four-banded linear systems. This decomposed Toeplitz system features multiple right-hand terms, significantly reducing computational complexity through Toeplitz inversion. Additionally, we introduced an algorithm based on banded LU decomposition, resulting in a banded linear system with multiple right-hand terms, where the sparsity of the banded LU decomposition is pivotal. To evaluate the algorithm's performance, we presented two examples in numerical simulations.