<i> G </i> -Subdiffusion Equation as an Anomalous Diffusion Equation Determined by the Time Evolution of the Mean Square Displacement of a Diffusing Molecule

Abstract

Normal and anomalous diffusion processes are characterized by the time evolution of the mean square displacement of a diffusing molecule $σ2(t)$. When $σ2(t)$ is a power function of time, the process is described by a fractional subdiffusion, fractional superdiffusion or normal diffusion equation. However, for other forms of $σ2(t)$, diffusion equations are often not defined. We show that to describe diffusion characterized by $σ2(t)$, the g-subdiffusion equation with the fractional Caputo derivative with respect to a function g can be used. Choosing an appropriate function g, we obtain Green’s function for this equation, which generates the assumed $σ2(t)$. A method for solving such an equation, based on the Laplace transform with respect to the function g, is also described.