G-Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function to Describe Ultraslow Diffusion

Ultraslow diffusion (slow subdiffusion) is usually defined as a random walk of a diffusing molecule such that its mean square displacement is controlled by a slowly varying function, in practice, by a combination of logarithmic functions. The Green’s function describing molecule diffusion is often determined in the long-time limit. We show that the g-subdiffusion equation with the fractional Caputo derivative with respect to another function g describes ultraslow diffusion in the entire time domain when the function g is appropriately chosen. To determine the Green’s function we use a method based on the Laplace transform with respect to the function g. We also use the g-subdiffusion equation to model a continuous transition between two different ultraslow diffusion processes.