# Fractional Diffusion to a Cantor Set in 2D

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## Abstract

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## 1. Introduction

#### Preliminaries: Subdiffusion and Boundary Conditions on a Comb

## 2. First Arrival Time Distribution for Subdiffusion

#### 2.1. Fractal Set of Sinks

#### 2.2. Single Sink

## 3. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. A Brief Survey on Fractional Integration

## Appendix B. Fractional Fokker–Planck Equation

#### Solution in the Form of the Fox Function

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**Figure 1.**A random walk on a comb in the presence of a fractal sink structure. This fractal structure, where the sink points are placed at $y=Y$, is shown in zoom. The comb consists of the backbone-x axis and the y direction corresponds to continuously distributed fingers along the backbone. This structure results from the phenomenological consideration of 2D diffusion with diffusion coefficients ${D}_{xx}={D}_{x}\delta \left(y\right)$ and ${D}_{yy}={D}_{y}$, where ${D}_{x},{D}_{y}$ are constant values [20].

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**MDPI and ACS Style**

Iomin, A.; Sandev, T.
Fractional Diffusion to a Cantor Set in 2D. *Fractal Fract.* **2020**, *4*, 52.
https://doi.org/10.3390/fractalfract4040052

**AMA Style**

Iomin A, Sandev T.
Fractional Diffusion to a Cantor Set in 2D. *Fractal and Fractional*. 2020; 4(4):52.
https://doi.org/10.3390/fractalfract4040052

**Chicago/Turabian Style**

Iomin, Alexander, and Trifce Sandev.
2020. "Fractional Diffusion to a Cantor Set in 2D" *Fractal and Fractional* 4, no. 4: 52.
https://doi.org/10.3390/fractalfract4040052