Time-fractional diffusion equation with time dependent diffusion coefficient

We consider the time-fractional diffusion equation with time dependent diffusion coefficient given by ${}_{0}O_{\left(C\right)t}^{\alpha }W(x,t)=D_{\alpha ,\gamma }{t}^{\gamma }[{\partial }^{2}W(x,t)∕\partial x^2]$, where ${}_{0}O_{\left(C\right)t}^{\alpha }$ is the Caputo operator. We investigate its solutions in the infinite and the finite domains. The mean squared displacement and the mean first passage time are also considered. In particular, for $\alpha =0$, the mean squared displacement is given by $⟨x^2⟩\sim t^{\gamma }$ and we verify that the mean first passage time is finite for superdiffusive regimes.