Dynamics of a Model of Polluted Lakes via Fractal-Fractional Operators with Two Different Numerical Algorithms

TL;DR

This paper models polluted lakes using fractal-fractional differential equations with Mittag-Leffler kernels, proving solution existence and stability, and simulates pollutant dynamics with two novel numerical algorithms under various input scenarios.

Contribution

It introduces a fractal-fractional model for polluted lakes with Mittag-Leffler kernels and develops two new numerical algorithms for simulation.

Findings

The model's solutions exist and are stable under certain conditions.

Numerical simulations demonstrate the impact of different pollutant input models.

Fractal-fractional derivatives significantly influence pollutant dynamics.

Abstract

We employ Mittag-Leffler type kernels to solve a system of fractional differential equations using fractal-fractional (FF) operators with two fractal and fractional orders. Using the notion of FF-derivatives with nonsingular and nonlocal fading memory, a model of three polluted lakes with one source of pollution is investigated. The properties of a non-decreasing and compact mapping are used in order to prove the existence of a solution for the FF-model of polluted lake system. For this purpose, the Leray-Schauder theorem is used. After exploring stability requirements in four versions, the proposed model of polluted lakes system is then simulated using two new numerical techniques based on Adams-Bashforth and Newton polynomials methods. The effect of fractal-fractional differentiation is illustrated numerically. Moreover, the effect of the FF-derivatives is shown under three specific input models of the pollutant: linear, exponentially decaying, and periodic.