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?
Some questions about fractional derivative
finite element;pde;space time galerkin