The propose of this paper is to introduce and investigate a highly accurate technique for solving the fractional Logistic and Ricatti differential equations of variable-order. We consider these models with the most common nonsingular Atangana–Baleanu–Caputo (ABC) fractional derivative which depends on the Mittag–Leffler kernel. The proposed numerical technique is based upon the fundamental theorem of the fractional calculus as well as the Lagrange polynomial interpolation. We satisfy the efficiency and the accuracy of the given procedure; and study the effect of the variation of the fractional-order [Formula: see text] on the behavior of the solutions due to the presence of ABC-operator by evaluating the solution with different values of [Formula: see text]. The results show that the given procedure is an easy and efficient tool to investigate the solution for such models. We compare the numerical solutions with the exact solution, thereby showing excellent agreement which we have found by applying the ABC-derivatives. We observe the chaotic solutions with some fractional-variable-order functions.

中文翻译：

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研究变阶ABC分数微分方程描述的一些混沌模型的高精度技术

本文的目的是介绍和研究一种求解变阶分数 Logistic 和 Ricatti 微分方程的高精度技术。我们用最常见的非奇异 Atangana-Baleanu-Caputo (ABC) 分数导数来考虑这些模型，它取决于 Mittag-Leffler 核。所提出的数值技术基于分数阶微积分的基本定理以及拉格朗日多项式插值。我们满足给定程序的效率和准确性；并通过评估具有不同值[公式：见文本]的解，研究由于存在 ABC 算子而导致的分数阶 [公式：见文本] 的变化对解的行为的影响。结果表明，给定的程序是研究此类模型的解决方案的一种简单有效的工具。我们将数值解与精确解进行比较，从而显示出我们通过应用 ABC 导数发现的极好的一致性。我们用一些分数变阶函数观察混沌解。