Beyond Classical Diffusion: Fractional Derivatives in Transport and Stochastic Systems

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.