Abstract In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.

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为什么不应该使用具有非奇异核的分数阶导数

摘要 近年来，许多论文讨论了通过用非奇异（即有界）核代替Caputo或Riemann-Liouville导数的奇异核构造的新分数阶导数的理论和应用。此处将通过严格的数学推理表明，这些非奇异核导数存在一些应禁止使用的缺点。他们不能满足分数阶微积分的基本定理，因为他们不承认存在相应的卷积积分，其导数是左逆的；并且在初始时间 t = 0 的导数值始终为零，这对可以使用这些导数的微分方程和模型施加了不自然的限制。