Composite subdiffusion equation that describes transient subdiffusion
Tadeusz Kosztołowicz, Aldona Dutkiewicz
Abstract
A composite 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 $\ensuremath{\alpha}$ and ${D}_{\ensuremath{\alpha}}$ to subdiffusion with parameters $\ensuremath{\beta}$ and ${D}_{\ensuremath{\beta}}$. The parameters are defined by the time evolution of the mean square displacement of diffusing particle ${\ensuremath{\sigma}}^{2}(t)=2{D}_{i}{t}^{i}/\mathrm{\ensuremath{\Gamma}}(1+i)$, $i=\ensuremath{\alpha},\ensuremath{\beta}$. The function $g$ controls the process at intermediate times. The composite subdiffusion equation is more general than the ordinary fractional subdiffusion equation with constant parameters; it has potentially wide application in modeling diffusion processes with changing parameters.