# From continuous-time random walks to controlled-diffusion reaction

## TL;DR

This paper develops an analytical model for anomalous diffusion involving coupled diffusion equations with localized reactions, extending it to include memory effects and connecting these to observed anomalous diffusion behaviors.

## Contribution

It introduces a novel coupled diffusion model with exact solutions and extends it to include memory effects, linking power-law kernels to anomalous diffusion.

## Key findings

- Exact analytical solutions for coupled diffusion with localized reactions.
- Extension of the model to include memory effects.
- Establishment of a relation between memory kernels and anomalous diffusion.

## Abstract

Daily, are reported systems in nature that present anomalous diffusion phenomena due to irregularities of medium, traps or reactions process. In this scenario, the diffusion with traps or localised--reactions emerge through various investigations that include numerical, analytical and experimental techniques. In this work, we construct a model which involves a coupling of two diffusion equations to approach the random walkers in a medium with localised reaction point (or controlled diffusion). We present the exact analytical solutions to the model. In the following, we obtain the survival probability and mean square displacement. Moreover, we extend the model to include memory effects in reaction points. Thereby, we found a simple relation that connects the power-law memory kernels with anomalous diffusion phenomena, i.e. $\langle (x-\langle x \rangle )^2 \rangle \propto t^{\mu}$. The investigations presented in this work uses recent mathematical techniques to introduces a form to represent the coupled random walks in context of reaction-diffusion problem to localised reaction.