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

Abstract

Within the framework of a new mathematical model of convective diffusion with the k-Caputo derivative, we simulate the dynamics of anomalous soluble substances migration under the conditions of two-dimensional steady-state plane-vertical filtration with a free surface. As a corresponding filtration scheme, we consider the scheme for the spread of pollution from rivers, canals, or storages of industrial wastes. On the base of a locally one-dimensional finite-difference scheme, we develop a numerical method for obtaining solutions of boundary value problem for fractional differential equation with k-Caputo derivative with respect to the time variable that describes the convective diffusion of salt solution. The results of numerical experiments on modeling the dynamics of the considered process are presented. The results that show an existence of a time lag in the process of diffusion field formation are presented.

Convergence Analysis of a Numerical Method for a Fractional Model of Fluid Flow in Fractured Porous Media

“…Similarly, we propose the following two fully discrete schemes that correspond to Cases II and III and are based on the semi-discrete schemes (20) and (21). Problem 9.…”

“…There are several definitions of the fractional derivative and fractional integral in fractional calculus, an overview of which can be found in [18,19]. In the development of mathematical models of fluid flow in porous media under different physical assumptions, various authors have used fractional time derivatives in the sense of Caputo [7][8][9]20,21], Riemann-Liouville [6,22], Caputo-Fabrizio [23,24], Atangana-Baleanu [25] and others [26] to account for memory effects, whereas the Riemann-Liouville derivative is mainly used to describe nonlocality in the spatial direction [6,27]. In the study of dynamic processes in fractured fractal media, a fractional temporal derivative in the sense of Caputo has an advantage, since it is given by the convolution of the power law kernel and the derivative of the function.…”

The Craft of Fractional Modelling in Science and Engineering: II and III

“…Real-world problems such as underground water filtration [8,9], fractional electronic elements [10], and viscous thermal losses in musical instruments (flutes) [11] show the power of fractional modelling in modeling complex phenomena with non-locality and energy dissipation.…”

Mathematical Modeling of Effect of Pumping Rate on Contaminant Transport in Riverbank Filtration System

“…Different parameters were varied and they observed a general decrease in contaminant concentration. Bohaienko and Bulavatsky (2019) developed a mathematical model of solutes migration under the conditions of groundwater filtration with k -Caputo fractional derivatives. They use finite difference scheme to simulate the dynamics of anomalous soluble substance migration under the conditions of twodimensional steady-state plane-vertical filtration with a free surface.…”