Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function in Modeling Superdiffusion.

Abstract

Superdiffusion is usually defined as a random walk process of a molecule, in which the time evolution of the mean-squared displacement, σ2, of the molecule is a power function of time, σ2(t)∼t2/γ, with γ∈(1,2). An equation with a Riesz-type fractional derivative of the order γ with respect to a spatial variable (a fractional superdiffusion equation) is often used to describe superdiffusion. However, this equation leads to the formula σ2(t)=κt2/γ with κ=∞, which, in practice, makes it impossible to define the parameter γ. Moreover, due to the nonlocal nature of this derivative, it is generally not possible to impose boundary conditions at a thin partially permeable membrane. We show a model of superdiffusion based on an equation in which there is a fractional Caputo time derivative with respect to another function, g; the spatial derivative is of the second order. By choosing the function in an appropriate way, we obtain the g-superdiffusion equation, in which Green's function (GF) in the long time limit approaches GF for the fractional superdiffusion equation. GF for the g-superdiffusion equation generates σ2 with finite κ. In addition, the boundary conditions at a thin membrane can be given in a similar way as for normal diffusion or subdiffusion. As an example, the filtration process generated by a partially permeable membrane in a superdiffusive medium is considered.