Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion

TL;DR

This paper introduces a generalized subdiffusion equation with a fractional Caputo derivative relative to another function, enabling modeling of a smooth transition from subdiffusion to superdiffusion over time.

Contribution

It develops a $g$--subdiffusion equation framework that captures the transition from subdiffusion to superdiffusion, using the $g$--Laplace transform and identifying the function $g$ for this transition.

Findings

The $g$--subdiffusion Green's function matches subdiffusion at short times and superdiffusion at long times.

The scaling properties of the Green's functions are similar in the long-time limit.

The model shows superdiffusive behavior can arise from increased jump frequency, not just long jumps.

Abstract

We use a subdiffusion equation with fractional Caputo time derivative with respect to another function $g$ ($g$--subdiffusion equation) to describe a smooth transition from ordinary subdiffusion to superdiffusion. Ordinary subdiffusion is described by the equation with the ``ordinary'' fractional Caputo time derivative, superdiffusion is described by the equation with a fractional Riesz type spatial derivative. We find the function $g$ for which the solution (Green's function, GF) to the $g$--subdiffusion equation takes the form of GF for ordinary subdiffusion in the limit of small time and GF for superdiffusion in the limit of long time. To solve the $g$--subdiffusion equation we use the $g$--Laplace transform method. It is shown that the scaling properties of the GF for $g$--subdiffusion and the GF for superdiffusion are the same in the long time limit. We conclude that for a sufficiently long time the $g$--subdiffusion equation describes superdiffusion well, despite a different stochastic interpretation of the processes. Then, paradoxically, a subdiffusion equation with a fractional time derivative describes superdiffusion. The superdiffusive effect is achieved here not by making anomalously long jumps by a diffusing particle, but by greatly increasing the particle jump frequency which is derived by means of the $g$--continuous time random walk model. The $g$--subdiffusion equation is shown to be quite general, it can be used in modeling of processes in which a kind of diffusion change continuously over time. In addition, some methods used in modeling of ordinary subdiffusion processes, such as the derivation of local boundary conditions at a thin partially permeable membrane, can be used to model $g$--subdiffusion processes, even if this process is interpreted as superdiffusion.