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

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.