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.
Highlights
We study the problem of modeling convective diffusion of solutes under the conditions of plane-vertical steady-state filtration with a free surface
The challenging issue is to increase the adequacy of classical quantitative models of heat and mass transfer in systems with a complex space-time structure characterized by memory effects, spatial non-locality, and self-organization
Significant progress while modeling convective diffusion in anomalous conditions and several other processes in the continuum mechanics was achieved using the fractional-order integro-differentiation formalism [5,6,7,8,9,10,11,12,13]. These transport processes are strongly non-local in time and space, mathematical models have the form of differential equations of fractional order
Summary
We study the problem of modeling convective diffusion of solutes under the conditions of plane-vertical steady-state filtration with a free surface. Significant progress while modeling convective diffusion in anomalous conditions and several other processes in the continuum mechanics was achieved using the fractional-order integro-differentiation formalism [5,6,7,8,9,10,11,12,13] These transport processes are strongly non-local in time and (or) space, mathematical models have the form of differential equations of fractional order. The methods most used in such case are finite-difference splitting schemes that reduce high-dimensional problems to a set of one-dimensional ones Such schemes for the classical models of diffusion were thoroughly studied in [14].
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