$G$-subdiffusion equation that describes transient subdiffusion

Abstract: A $g$--subdiffusion equation with fractional Caputo time derivative with respect to another function $g$ is used to describe a process of a continuous transition from subdiffusion with parameters $\alpha$ and $D_\alpha$ to subdiffusion with parameters $\beta$ and $D_\beta$. The parameters are defined by the time evolution of the mean square displacement of diffusing particle $\sigma^2(t)=2D_i t^i/\Gamma(1+i)$, $i=\alpha,\beta$. The function $g$ controls the process at "intermediate" times. The $g$--subdiffusion equation is more general than the "ordinary" fractional subdiffusion equation with constant parameters, it has potentially wide application in modelling diffusion processes with changing parameters.