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Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition
1Institute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, Poland.
Physical Review. E
|July 19, 2023
Summary
This study introduces a g-subdiffusion equation that smoothly transitions between subdiffusion and superdiffusion. The g-subdiffusion model effectively describes superdiffusion over long times, even with fractional time derivatives, by increasing particle jump frequency.
Area of Science:
- Mathematical Physics
- Statistical Mechanics
- Complex Systems
Background:
- Subdiffusion and superdiffusion are anomalous diffusion processes crucial in various scientific fields.
- Existing models often describe either subdiffusion or superdiffusion distinctly.
- A unified framework for transitioning between these regimes is lacking.
Purpose of the Study:
- To introduce and analyze a g-subdiffusion equation with a fractional Caputo time derivative with respect to a function g.
- To demonstrate a smooth transition from ordinary subdiffusion to superdiffusion within a single model.
- To find the specific function g enabling this transition in the Green's function behavior.
Main Methods:
- Utilizing a g-subdiffusion equation with a fractional Caputo time derivative with respect to a function g.
- Employing the g-Laplace transform method for solving the g-subdiffusion equation.
- Analyzing the Green's function (GF) behavior in small and long time limits.
Main Results:
- The g-subdiffusion equation allows for a smooth transition from ordinary subdiffusion to superdiffusion.
- The Green's function of the g-subdiffusion equation mimics ordinary subdiffusion at short times and superdiffusion at long times.
- The scaling properties of the g-subdiffusion Green's function match superdiffusion in the long-time limit.
- Superdiffusion is achieved by increasing particle jump frequency via the g-continuous-time random walk model, not by longer jumps.
Conclusions:
- The g-subdiffusion equation provides a versatile model for processes exhibiting time-dependent diffusion changes.
- For extended durations, the g-subdiffusion equation accurately models superdiffusion, despite its fractional time derivative formulation.
- The model offers a paradoxical yet effective way to describe superdiffusion using subdiffusion-based equations.
- Established methods for subdiffusion modeling can be adapted for g-subdiffusion processes, including superdiffusive interpretations.
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