# Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the k-Caputo Fractional Derivative

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and the Corresponding Boundary Value Problem

## 3. Approximate Solution of the Boundary Value Problem

## 4. Results of Numerical Experiments

- There is a delay in the formation of concentration field while modeling the diffusion process using the model with the k-Caputo derivative comparing both to the case of modeling this process using the classical model [1,2] and the model based on the standard definition of the Caputo derivative (Figure 3).
- For fixed values of the parameter k, when the value of $\alpha $ varies, there is also a delay in the advance of concentration front for the diffusion model with the k-Caputo derivative compared with the case of the classical model. This delay is the greater, the smaller the value of the parameter $\alpha $ is (curves 2–4 in Figure 5). The performed computations also allow us to conclude that the dynamics of concentration front’s delay is determined to a greater extent by a change in the magnitude of order $\alpha $ than by a change of the value of the parameter k.

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Bohaienko, V.; Bulavatsky, V.
Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the *k*-Caputo Fractional Derivative. *Fractal Fract.* **2018**, *2*, 28.
https://doi.org/10.3390/fractalfract2040028

**AMA Style**

Bohaienko V, Bulavatsky V.
Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the *k*-Caputo Fractional Derivative. *Fractal and Fractional*. 2018; 2(4):28.
https://doi.org/10.3390/fractalfract2040028

**Chicago/Turabian Style**

Bohaienko, Vsevolod, and Volodymyr Bulavatsky.
2018. "Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the *k*-Caputo Fractional Derivative" *Fractal and Fractional* 2, no. 4: 28.
https://doi.org/10.3390/fractalfract2040028