Fractional Diffusion Equations: An Overview

Title: Fractional Diffusion Equations: An Overview

Research Question: How can we solve fractional diffusion equations, which describe anomalous transport phenomena, using numerical methods?

Methodology: We propose an explicit finite difference method for solving fractional diffusion equations. This method combines the forward time centered space (FTCS) method, known for integrating ordinary diffusion equations, with the Grünwald-Letnikov definition of the fractional derivative operator. This results in an explicit FTCS scheme for solving fractional diffusion equations.

Results: We present a stability analysis using von Neumann methods, showing that the analytical stability bounds are in agreement with numerical tests. We compare our numerical predictions with exact analytical solutions and find good agreement.

Implications: Our method provides a practical approach to solving fractional diffusion equations, which have wide-ranging applications in fields such as physics, chemistry, and geology. This could lead to advancements in our understanding and modeling of anomalous transport phenomena.

Link to Article: https://arxiv.org/abs/0311011v1 Authors: arXiv ID: 0311011v1