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

4. Findings and discussion.
For comparison, the thermal conductivity equations are used here as a test example and graphical resultsare obtained. In contrast to the heat transfer equation, the salt transfer equation contains terms of low pro derivatives. In order toget rid of these terms, and to reduce this equation to the form of a heat equation,a replacement of the variablex is made. After that, the methods proposed by Samara Sobol were chosen to solve the problem. To solve the one-dimensional problem, the iteration method is used, and for each iteration, the resulting system of algebraic equations with a tridiagonal matrix is solved by the run method. When solving a two-dimensional problem, the method of variable directions is used, i.e. calculations are performed according to the Pisman –Ruckford scheme. When using this scheme, a two-dimensional problem is reduced to two one-dimensional problems, which are solved in the above way. The main results obtained in the article are::

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  Estimates of the behavior of the free boundary for the convective salt transfer equation in a medium with variable permeability, which are important for numerical modeling, are constructed.
  
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  Estimates and asymptotics of solutions are obtained, including for the case of strong absorption.
  
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  The asymptotic behavior of solutions to the Cauchy problem at critical values of parameters is investigated.
  
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  Convergence of self-similar solutions to solutions of the Cauchy problem is proved.
  
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  New nonlinear effects are obtained.
  
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  Numerical schemes of algorithms and programs for assigned tasks are developed in the universal mathematical environment Matlab.
  

5. Conclusion.
The use of moderncomputersystems for solving problems of mathematical physics, in particular for the problem of the process of water and salt transfer in soils based on models of the movement of multicomponent impurities in porous media has become the most urgent task. The study of various properties of self-similar (one-dimensional) equations, which are simpler than multidimensional partial differential equations, is a relatively easy task and therefore equations of this kind lend themselves to a more detailed analysis[6]-[7],[9],[10]. The obtained numerical results are used to construct graphical results of the problem in animated form. The problem was considered in a one-dimensional and two-dimensional domain, and, accordingly, the graphical results are shown in two-dimensional space.