Compounded random walk for space-fractional diffusion on finite domains

Area of Science:

• Physics
• Applied Mathematics
• Computational Science

Background:

• Fractional diffusion models are crucial for describing anomalous transport phenomena.
• Existing models often face limitations on bounded domains or when incorporating external forces.
• Understanding diffusion in complex environments requires robust mathematical frameworks.

Purpose of the Study:

• To introduce a novel compounded random walk formulation.
• To establish its connection with fractional Fokker-Planck and diffusion equations.
• To provide a versatile tool for modeling diffusion in bounded and unbounded spaces with forces.

Main Methods:

• Formulation of a compounded random walk with space-dependent forces.
• Derivation of the governing evolution equation.
• Analysis of its limiting behavior on finite and infinite domains.
• Development of numerical approximation and Monte Carlo simulation techniques.

Main Results:

• The compounded random walk is physically well-defined on finite and infinite domains.
• Its evolution equation limits to a space-fractional Fokker-Planck equation on bounded domains.
• It recovers the superdiffusive space-fractional diffusion equation on infinite domains.
• Numerical simulations show excellent agreement with analytical solutions.

Conclusions:

• The compounded random walk offers a significant advancement for modeling space-fractional diffusion.
• It provides a unified framework for diffusion on finite domains and in potential fields.
• This approach enhances the capability to simulate complex transport phenomena.