关于用非singular kernel定义分数阶导数的错误
On the mistake in defining fractional derivative using a non-singular kernel
摘要：
最近提出了使用非奇异核的分数阶导数$α$（$0 < α\leq 1$）的定义。在本说明中，我们表明，这些定义不能是有用的建模问题的初值条件（如，例如，分数扩散方程），因为这些方程得到的解决方案不满足初始条件（除了整数情况$α= 1$）。为了满足任意的初始条件，分数阶导数的定义必须包含一个奇异核。
英文摘要：
Definitions of fractional derivative of order $α$ ($0 < α\leq 1$) using non-singular kernels have been recently proposed. In this note we show that these definitions cannot be useful in modelling problems with a initial value condition (like, for example, the fractional diffusion equation) because the solutions obtained for these equations do not satisfy the initial condition (except for the integer case $α= 1$). In order to satisfy an arbitrary initial condition the definitions of fractional derivative must necessarily involve a singular kernel.
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