Abstract To tackle several limitations recently highlighted in the field of fractional differentiation and fractional models, some authors have proposed new kernels for the definition of fractional integration/differentiation operators. These kernels still have some limitations, however. One goal of this paper is thus to propose other kernels with a power law type behavior that overcome these limitations. All these kernels are nonsingular and have a limited memory. They are then used to define a class of models adapted to capture input-output power law type long memory behaviors. The stability of this class of model is investigated. Finally, the paper shows that the fractional pseudo state space description, a fractional model widely used in the literature, is a special case of Volterra equations, equations introduced nearly a century ago. Volterra equations, whose memory length can be explicitly controlled, can thus be viewed as a serious alternative to fractional pseudo state space descriptions for power law type long memory behavior modeling, as fractional pseudo state space descriptions are known to exhibit serious drawbacks also discussed in the paper.

中文翻译：

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用于建模幂律类型长内存行为及其他行为的非奇异内核

摘要 为了解决分数微分和分数模型领域最近突出的几个限制，一些作者提出了用于定义分数积分/微分算子的新内核。然而，这些内核仍然有一些限制。因此，本文的一个目标是提出具有幂律类型行为的其他内核，以克服这些限制。所有这些内核都是非奇异的，并且内存有限。然后，它们用于定义一类模型，适用于捕获输入-输出幂律类型的长记忆行为。研究了此类模型的稳定性。最后，论文表明分数伪状态空间描述是文献中广泛使用的分数模型，是近一个世纪前引入的沃尔泰拉方程的特例。