论文标题
为什么不使用非词内核的分数衍生物
Why fractional derivatives with nonsingular kernels should not be used
论文作者
论文摘要
近年来，许多论文讨论了通过取代Caputo或Riemann-Liouville衍生物的奇异阶数衍生物的理论和应用，这些衍生物是通过非单星（即有界）内核来构建的。这里将通过严格的数学推理来显示这些非单明核衍生物遭受的几个缺点，这些缺点应该禁止使用它们。他们无法满足分数演算的基本定理，因为他们不承认存在相应的卷积积分，该卷积积分是衍生物是剩余的。并且在初始时间$ t = 0 $的衍生物的值始终为零，这对可以使用这些衍生物的微分方程和模型对差异方程式和模型施加了不自然的限制。对于所谓的Caputo-Fabrizio和Atangana-Baleanu衍生物的特殊情况，可以简单地以整数衍生物和标准的CAPUTO分数衍生物来表达衍生物时，表明这些衍生物可以表明这些衍生物没有任何新的东西。
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.
