Fractal–Fractional Operators Applied to Water Pollution Model: Well Posedness, Stability, and Simulation

Water contamination is a crucial area of study that has drawn significant attention from researchers and environmentalists due to its profound impact on humans, animals, and plants. It is equally harmful as air and soil contamination and is closely linked to both. Due to the global importance of this issue, accurately analyzing mathematical models is one of the priorities for researchers. In this regard, this paper presents a mathematical model for both soluble and insoluble water pollutants, formulated as a system of nonlinear fractional ordinary differential equations using a compartmental approach. The fractal–fractional (FF) derivative operator with power, exponential, and generalized Mittag–Leffler kernels is employed to solve the proposed system. The existence and uniqueness of the FF order model are proven using Schaefer’s fixed-point theorem and the Banach contraction principle. Additionally, we demonstrate global stability based on the Ulam–Hyers approach. Finally, simulations have been carried out using the Adams–Bashforth techniques to derive a fractional-order scheme that will help to validate the theoretical findings and provide insights into how to prevent and control this type of environmental problem.