On the issue of mathematical modeling of salt transfer processes in soils taking into account convectional transference Abstract

1. Problem statement. In a liquid with a concentration moving at a velocity parallel to the axis , diffusion takes place that is nonlinear to the equations with a free boundary, in the region of 
(1)
(2)
where is the salt concentration at time and at a point _, and is the convective transport rate at time and at a point . - the diffusion coefficient and . Here is the nonlinearity parameter, whose values are different for different physical phenomena and which characterizes the nonlinearity of the medium.
and - are functions of quantities that depend on the type of soil, soil structure of pore space, soil moisture, and other factors, describing the process of salt transfer in a nonlinear medium, in the presence of convective transport with velocity . We consider cases when , and .
In the case when upper solutions of the problem (1)-(2) with a free boundary are found, assuming that . A visualization of the nonlinear salt transfer process described by problem (1)-(2) is presented, taking into account convective transport with velocity . In the case when a number of self-similar solutions were constructed in [3-4] and a numerical analysis of the solution is given. And semichen newyy non linearyy effectt Based on these effects, for the case when upper and lower bounds and localization conditions for solutions are obtained. The asymptotic behavior of solutions near the free boundary is obtained by the methods used in [5]-[6]. Numerical calculations are performed based on the established properties of the problem solutions. 
Visualization of a nonlinear process described by quasi-linear equations requires finding out the nature of solutions depending on the value of the parameters included in the original equation. Results and visualization of solutions obtained using the capabilities of the Matlab program.
In the study ofnonlinear problems, theorems of comparison of solutions play an important role. Having found a partial solution of a self-similar or approximate self-similar equation, you can then use it to compare solutions, which makes it possible, without knowing the solution of the problem, to get an estimate of the solutions through a known function, which is very importantfor numerical solutions ofnonlinear problems.
The theorem.1. Let b dene a non-negative generalized solution of problem (3)-(4) and functions , where are continuous functions satisfying the inequalities in and 
Then the estimates are valid
(sup),
(sub).
The functions are called upper and lower solutions of problem (1)-(2), respectively.