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I have read some papers about time-fractional PDEs solved by finite element method. The time-fractional derivative is the Caputo derivative defined by $$ \frac{\partial^{\alpha}u}{\partial t^{\alpha}}=\frac{1}{\Gamma(1-\alpha)}\int_{0}^t\frac{\partial u}{\partial s}\frac{ds}{(t-s)^{\alpha}},~~~~0<\alpha<1. $$

Now, we consider the following time-fractional diffusion equation \begin{align} &\frac{\partial^{\alpha}u}{\partial t^{\alpha}}-\Delta u=f,~~in~~\Omega,\\ &u=0,~~~~~~~~~on~~~\partial\Omega,\\ &u(0)=u_0, \end{align} where $\Omega$ is a convex and bounded domain.

1.What does mean the singularity of the time-fractional derivative?

2.In what cases, the singularity of the time-fractional derivative can lead to low smoothness of solution $u$.

3.What does mean the regularity of a function is not smooth about time?

Could anyone give me some explanations?

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  • $\begingroup$ I don't understand what you have written. Are you claiming that the fractional derivative is singular? $\endgroup$ – David Ketcheson Mar 2 '17 at 21:34
  • $\begingroup$ Yes, how to understand the singularity of the time-fractional derivative?@DavidKetcheson $\endgroup$ – Feng Young Mar 3 '17 at 0:51
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    $\begingroup$ What @DavidKetcheson is asking is "what do you mean when you say that the time-fractional derivative is singular"? I assume you read this statement somewhere, but it is not clear what it is supposed to mean without the context where you found it. $\endgroup$ – Wolfgang Bangerth Mar 3 '17 at 4:45

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