Cattaneo--type subdiffusion equation

The ordinary subdiffusion equation, with fractional time derivatives of at most first order, describes a process in which the propagation velocity of diffusing molecules is unlimited. To avoid this non-physical property the Cattaneo diffusion equation has been proposed. Compared to the ordinary subdiffusion equation, the Cattaneo equation contains an additional time derivative of order greater than one and less than or equal to two. The fractional order of the additional derivative depends on the subdiffusion exponent. We study a Cattaneo-type subdiffusion equation (CTSE) that differs from the ordinary subdiffusion equation by an additional integro--differential operator (AO) which may be independent of subdiffusion parameters. The AO describes processes affecting ordinary subdiffusion. The equation is derived combining the modified diffusive flux equation with the continuity equation. It can also be obtained within the continuous time random walk model with the waiting time distribution for the molecule to jump controlled by the kernel of AO. We discuss whether the ordinary subdiffusion equation and CTSE provide qualitative differences in the description of subdiffusion. For example, we consider two types of CTSE: the CTSE with AO which is the Caputo fractional time derivative of the order independent of the subdiffusion exponent and with the AO with a kernel that is a slowly varying function. In the first case the effect generated by AO disappears relatively quickly over time. In the second one the effect may be visible for a long time.