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

Area of Science:

• Physics
• Physical Chemistry
• Mathematical Modeling

Background:

• Superdiffusion describes molecular random walks where mean-squared displacement scales as σ2(t)∼t2/γ, with γ∈(1,2).
• Traditional fractional superdiffusion equations use spatial Riesz derivatives, leading to infinite parameters (κ=∞) and difficulties with boundary conditions at membranes.

Purpose of the Study:

• To introduce a novel g-superdiffusion model addressing limitations of existing fractional superdiffusion equations.
• To enable finite parameter (κ) estimation and facilitate boundary condition implementation at thin membranes.

Main Methods:

• Developed a superdiffusion model utilizing a fractional Caputo time derivative with respect to a function 'g' and a second-order spatial derivative.
• Analyzed the Green's function (GF) of the g-superdiffusion equation, showing its long-time limit approximates that of fractional superdiffusion.
• Demonstrated the GF yields a finite κ for mean-squared displacement.

Main Results:

• The g-superdiffusion equation yields a finite κ, allowing for practical parameter γ determination.
• The model successfully accommodates boundary conditions at thin membranes, analogous to normal or subdiffusion scenarios.
• The Green's function for the g-superdiffusion equation approximates that of fractional superdiffusion in the long-time limit.

Conclusions:

• The g-superdiffusion model provides a mathematically tractable and physically relevant framework for studying superdiffusion phenomena.
• This model overcomes key limitations of traditional fractional superdiffusion equations, particularly regarding parameter definition and boundary condition application.
• The model is applicable to phenomena like filtration in superdiffusive media involving partially permeable membranes.