Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Beyond Classical Diffusion: Fractional Derivatives in Transport and Stochastic Systems (2503.13096v1)

Published 17 Mar 2025 in math.DS and math.PR

Abstract: Integer-order differential operators were originally used to describe local and isotropic effects, in both space and time. However, in fields like biology, the modelling of complex phenomena with spatial heterogeneity necessitates more advanced approaches. The fractional calculus framework provides powerful tools for developing models that better capture the intricate dynamics of biological systems. This paper derives fractional reaction-diffusion equations from continuous-time random walks, highlighting the role of heavy-tailed distributions in the process. Both fractional partial differential equations, on the macroscopic level, as well as fractional stochastic differential equations, on the microscopic level, will be derived and simulated from, for simple Riesz-fractional diffusion models. A new numerical scheme that implements periodic boundary conditions is proposed to control the loss of mass density. We highlight the key differences between fractional and classical diffusion.

Summary

We haven't generated a summary for this paper yet.