Abstract Humans are part of nature, and as nature existed before mankind, mathematics was created by humans with the main aim to analyze, understand and predict behaviors observed in nature. However, besides this aspect, mathematicians have introduced some laws helping them to obtain some theoretical results that may not have physical meaning or even a representation in nature. This is also the case in the field of fractional calculus in which the main aim was to capture more complex processes observed in nature. Some laws were imposed and some operators were misused, such as, for example, the Riemann–Liouville and Caputo derivatives that are power-law-based derivatives and have been used to model problems with no power law process. To solve this problem, new differential operators depicting different processes were introduced. This article aims to clarify some misunderstandings about the use of fractional differential and integral operators with non-singular kernels. Additionally, we suggest some numerical discretizations for the new differential operators to be used when dealing with initial value problems. Applications of some nature processes are provided.

中文翻译：

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对没有奇异核的微分算子的一些误解和缺乏理解

摘要 人类是自然界的一部分，由于自然界先于人类存在，因此人类创造了数学，其主要目的是分析、理解和预测在自然界中观察到的行为。然而，除此之外，数学家还引入了一些定律，帮助他们获得一些可能没有物理意义甚至在自然界中没有表示的理论结果。在分数阶微积分领域也是如此，其中的主要目标是捕捉自然界中观察到的更复杂的过程。一些定律被强加，一些算子被滥用，例如，黎曼-刘维尔和卡普托导数是基于幂律的导数，并已被用于对没有幂律过程的问题进行建模。为了解决这个问题，引入了描述不同过程的新微分算子。本文旨在澄清一些关于非奇异核使用分数阶微分和积分运算符的误解。此外，我们建议在处理初值问题时对新的微分算子进行一些数值离散化。提供了一些自然过程的应用。