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

3. Self-similar and approximately self-similar problem solution
We consider two cases separately. Let us construct self-similar and approximately self-similar equations for (1) - (2). Studying the properties of solutions, which are much easier than studying the properties of solutions to the original equation. For this purpose, we use the nonlinear splitting method proposed in [3-4].
Case I .
Lemma 1. Let and consider the Cauchy problem (1)-(2) in . Equation (1) has a self-similar solutionof the followingform
,, where (3)
here  , , and the function
(4)
b satisfiesthe self-similar equation
(5)
where is a constant, and (4) is an upper solution of problem (1)-(2) if , .
Proof. In fact, we calculate Then given (3),(4),(5) we have in the area of 
where . From the solution comparison theorem, it is easy to calculate that in . The lemma is proved.
Case II. To obtain a solution forthe stationary case in equation (1), we replace 

(6)
To study the latter equation, we use Hardy's theorem.

Theorem 2. (Hardy). Let be a function from the Hardy class. Then, for sufficiently large numbers, the asymptotic representation is valid 
Substituting this representation in (6), we have 
and integrate 

(7)
(7) is the solution of problem (1)-(2) for .

4. Difference schemes for solving two-dimensional salt transfer equations
For an approximate solution of problem (1)-(2), we construct a difference grid And a temporary grid 
We replace problem (1)-(2) with an implicit difference scheme using the balance method and obtain a difference problem with an error of :
(8)
Here or 
To solve the system of nonlinear equations, we apply various iterative methods and obtain:
(9)
Now the difference scheme (9) is relatively linear [5].
We introduce the following notation:
, , ,
For the numerical solution of the system of equations (7), we apply the run-through method [5].
In the course of numerical solution of the problem (1)-(2) for the cases of one-dimensional
and two-dimensional computational experiments were performed. Provided by
several results of a numerical experiment.

Figure 1. Distribution of the salt flow at the initial and final time points for the value when the initial time point is .

2. The diffusion coefficient increases.