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This paper discusses the calculation of distribution of hydraulic head H (x, y), [m] in flux of ground water, which is described by the known steady state groundwater flow equation with nonlinear boundary condition at unknown surface boundary. A special iterative algorithm for a numerical solution of the finite-difference approximation of the above-stated problem using known Richardson's over-relaxation method has been developed. The algorithm for parallel calculation is developed using message passing interface (MPI) for two dimensional case. It has been shown by calculation that above insertion with small coefficient of filtration, a local depression of water surface has been formed. This is the area without outflow of ground water. It may substantially cause the growth of soil salinization above surface of that depression due to evaporation of ground water, as dissolved salt stays in soil layer. Dependencies of the process on evaporation rate and depth of insertion are also discussed. An approximate 58% efficiency in time was obtained using four processes in cluster using message passing interface.

In this paper, the boundary value problem for the heat equation in the region which degenerates at the initial time is considered. Such problems arise in mathematical models of the processes occurring by opening of electric contacts, in particular, at the description of the heat transfer in a liquid metal bridge and electric arcing. The boundary value problem is reduced to a Volterra integral equation of the second kind which has a singular feature. The class of solutions for the integral equation is defined and the constructive method of its solution is developed.

The paper presents the results of modeling of the processes of phases transformations occurring in cathode of plasmatron with zirconium insertion. Model describes temperature and liquid-solid phase transformation in cathode considering kinetics of transformation in accordance with a state diagram. The comparison between one-dimensional mathematical models was exploited for estimation of the kinetics coefficient. First model is based on well-known heat equation with Stefans condition on the free boundary between liquid and solid phases [. The standard analytical self-similar solution for two-phase case is applied. In the second model, for heat equation instead of Stefans conditions, differential equations of kinetics are used.

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