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.

中文翻译：

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使用两种不同数值算法的分形-分数算子的污染湖泊模型的动力学

我们采用 Mittag-Leffler 型核来使用具有两个分形和分数阶的分形-分数 (FF) 算子来求解分数阶微分方程组。使用具有非奇异和非局部衰落记忆的 FF 导数概念，研究了具有一个污染源的三个污染湖泊的模型。利用非减紧映射的性质证明了污染湖泊系统FF模型解的存在性。为此，使用了 Leray-Schauder 定理。在探索了四个版本的稳定性要求后，使用基于 Adams-Bashforth 和牛顿多项式方法的两种新数值技术对所提出的污染湖泊系统模型进行了模拟。分形-分数微分的效果用数字说明。此外，FF 导数的效果在污染物的三种特定输入模型下显示：线性、指数衰减和周期性。