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

On the issue of mathematical modeling of salt transfer processes in soils taking into account convectional transference
Abstract. The article is devoted to the problems of mathematical modeling of the motion of matter in porous inhomogeneous media in relation to the transport of salts in soils. The need for mathematical modeling of such processes is determined both by the importance of the salt regime of soils for agricultural production, and by the effectiveness of mathematical modeling as a tool for understanding natural phenomena. For comparison, the thermal conductivity equations were chosen as a test example and the graphical results were obtained. Important estimates of the behavior of the free boundary for a nonlinear salt transfer equation with account for convective transport are constructed for numerical modeling. Numerical schemes of algorithms for the given problem are developed.

Key words. Salt regime, convective transport, self-similar equation, difference schemes.
1. Introduction. The salt regime is an important land reclamation regime that determines soil fertility, soil structure, plant growth conditions and development. Regulation of the salt regime is an important task of land reclamation and involves the use of technologies based on modern achievements of science and technology for these purposes.
In [1], the general solution of boundary value problems is studied for salt transfer problems and sufficient conditions for the existence of optimal controls (coefficients and generalized right-hand sides) and controllability are obtained. The necessary conditions for the extremum are obtained explicitly and a well-known scheme is used, which is based on solving direct and conjugate boundary value problems.
The problem of combating salinization of irrigated land has existed since the emergence of irrigation and, despite centuries of experience, remains relevant today.
The main reason for salinization of soils on irrigated and developed lands is the rise of the ground water level[8]. As an active means of combating salinization of land, their washing against the background of drainage acts. In this regard, such work will be carried out in the coming years on an area of several thousand hectares. It is obvious that the development and justification of reliable methods of salinity control for various natural and economic conditions are of considerable scientific interest and are of great practical importance.
However, a huge number of ongoing and even more forthcoming works on salinization of irrigated land in various natural and economic conditions require a more in-depth study of this problem and more reliable practical recommendations.
Technical, ecological, economic and other systems studied by modern science are no longer presented to research in the required completeness and accuracy by conventional theoretical methods. Computational experiments with mathematical models of objects allow you to study objects in detail and in depth in sufficient completeness, relying on modern computational algorithms. Direct natural experiment is expensive and long. Therefore, working not with the object itself, but with its model makes it possible to painlessly, relatively quickly and without significant costs, investigate its properties and behavior in any possible situations. situations.
The main problems that arise in the study of complex real physical processes are primarily related to the nonlinearity of the equations underlying the mathematical model. It is possible to obtain solutions of nonlinear boundary value problems in an analytical form, especially in multidimensional cases, only in exceptional cases. Therefore, they resort to various approximate methods for solving nonlinear problems, which is quite widely presented in [1-5]. In the works of many scientists, it is shown that one of the most effective methods for studying the properties of solutions of quasi-linear equations of parabolic type is self-similar and approximate self-similar approach and methods of comparing solutions: A. S. Kalashnikov was the first to study the technique of comparing solutions, thanks to which he was able to establish new properties of solutions to the Cauchy problem for quasilinear parabolic equations of degenerate type.
To optimize land irrigation regimes, it is necessary to use mathematical methods and appropriate software.
After stopping irrigation and with intensive evaporation from the soil surface, the salt concentration in the solution increases in the root layer (salt deposition in the pores is also possible). [2]
It is obvious that the concentration of soil solution in the aeration zone will also change depending on the irrigation regimes and norms. From the point of view of rational use of land water resources in the Aral Sea region, it is necessary to set an irrigation regime in which plants will receive sufficient moisture, and secondary salinity will be minimal. For some natural conditions, these requirements may not be met simultaneously. In this case, it is necessary to proceed from the condition that the salt concentration in the solution does not exceed the concentration at which plant growth is inhibited.
In [1-2], the general solution of boundary value problems is studied for salt transfer problems and sufficient conditions for the existence of optimal controls (coefficients and generalized right-hand sides) and controllability are obtained. The necessary conditions for the extremum are obtained explicitly.
The mathematical model of the distribution and transport of matter in porous media is based on the hypothesis that the movement of salts in a liquid can be considered as forced convection accompanied by diffusion. Many researchers have drawn attention to the importance of mathematical modeling of the processes of moisture and soluble salts entering the soil, their redistribution, consumption, and joint movement [1-15].
In this problem, the main problem is the non - linearity of the mathematical model, and they cannot be solved by classical methods. The paper deals with the nonlinear salt transfer problem, and self-similar and approximately self-similar solutions are obtained [3].