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

Anomalous and normal 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 fractional subdiffusion, fractional superdiffusion or normal diffusion equation. However, for other forms of σ2(t), the 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 the 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.