The work is dedicated to mathematical modelling of random diffusion flows of admixture particles in a two-phase stratified strip with stochastic disposition of phases and random thickness of inclusion-layers. The study of such models are especially important during the creation of composite layered materials, in the research of the transmission properties of filters, and in the prediction of the spread of pollutants in the environment. Within the model we consider one case of uniform distribution of coordinates of upper boundaries of the layers of which the body is made up and two more cases, i.e., of uniform and triangular distributions of the inclusion thickness. The initial-boundary value problems of diffusion are formulated for flux functions; the boundary conditions at one of the body’s surfaces are set for flux and, at the other boundary, the conditions are given for admixture concentration; the initial condition being concerned with zero and non-zero constant initial concentrations. An equivalent integro-differential equation is constructed. Its solution is found in terms of Neumann series. For the first time it was obtained calculation formulae for diffusion flux averaged over the ensemble of phase configurations and over the inclusion thickness. It allowed to investigate the dependence of averaged diffusion fluxes on the medium’s characteristics on the basis of the developed software. The simulation of averaged fluxes of admixture in multilayered $Fe-Cu$ and $\alpha Fe-Ni$ materials is made. Comparative analysis of solutions, depending on the stage of averaging procedure over thickness, is carried out. It is shown that for some values of parameters the stage of averaging procedure over thickness has almost no effect on the diffusion flow value.

We propose and justify a numerical method for computing the double integral with variable upper limits that leads to the variableness of the region of integration. Imposition of simple variables as functions for upper limits provides the form of triangles of integration region and variable in the external limit of integral leads to a continuous set of similar triangles. A variable grid is overlaid on the integration region. We consider three cases of changes of the grid for the division of the integration region into elementary volumes. The first is only the size of the imposed grid changes with the change of variable of the external upper limit. The second case is the number of division elements changes with the change of the external upper limit variable. In the third case, the grid size and the number of division elements change after fixing their multiplication. In these cases, the formulas for computing double integrals are obtained based on the application of cubatures in the internal region of integration and performing triangulation division along the variable boundary. The error of the method is determined by expanding the double integral into the Taylor series using Barrow’s theorem. Test of efficiency and reliability of the obtained formulas of the numerical method for three cases of ways of the division of integration region is carried out on examples of the double integration of sufficiently simple functions. Analysis of the obtained results shows that the smallest absolute and relative errors are obtained in the case of an increase of the number of division elements changes when the increase of variable of the external upper limit and the grid size is fixed.

In this work, a mathematical model of the transfer process in a layer under the condition of given experimental data on a part of the layer boundary is presented and investigated. Such research is important for the mathematical description of the objects and systems for which, based on physical considerations, it is impossible to correctly impose boundary or initial conditions, even in a sufficiently general form, but there are experimental data on the desired function or its derivative at the boundary of the body or at the initial time. The values of the desired function at the boundary are known at certain moments in time. The boundary condition is constructed by the experimental data and the initial-boundary value problem, with such a boundary condition, is formulated and solved. The influence of the statistical characteristics of the sample of experimental data on the solution to the initial-boundary value problem is analyzed, and a two-sided statistical estimation of the solution is determined. The confidence intervals for the coefficients of the regression equation and the corresponding confidence intervals for the sought function are established. The influence of the statistical characteristics of the sample on the sought function at the lower boundary of the layer is investigated. Numerical analysis of the solution to the initial-boundary value problem is carried out depending on the statistical characteristics of the sample. Various cases of samples by size and variance are considered. Numerical solutions are studied under the conditions of large and small time intervals of the considered process.

Based on the thermodynamic approach, a mathematical model of chemical water purification in a mechanical filter is constructed. For each of the components of the system, the mass transfer equation is obtained, on the basis of which the initial-boundary value problem of convective diffusion accompanied by a chemical reaction is formulated. Considering the terms corresponding to the components generated as a result of chemical reactions as internal sources, the system of integro-differential equations is assigned with the original initial-boundary value problem. The solution of this system is found by the method of successive iterations.

Modeling of impurity diffusion processes in a multiphase randomly inhomogeneous body is performed using the Feynman diagram technique. The impurity diffusion equations are formulated for each of the phases separately. Their random boundaries are subject to non-ideal contact conditions for concentration. The contact mass transfer problem is reduced to a partial differential equation describing diffusion in the body as a whole, which accounts for jump discontinuities in the searched function as well as in its derivative at the stochastic interfaces. The obtained problem is transformed into an integro-differential equation involving a random kernel, whose solution is constructed as a Neumann series. Averaging over the ensemble of phase configurations is performed. The Feynman diagram technique is developed to investigate the processes described by parabolic partial differential equations. The mass operator kernel is constructed as a sum of strongly connected diagrams. An integro-differential Dyson equation is obtained for the concentration field. In the Bourret approximation, the Dyson equation is specified for a multiphase randomly inhomogeneous medium with uniform phase distribution. The problem solution, obtained using Feynman diagrams, is compared with the solutions of diffusion problems for a homogeneous layer, one having the coefficients of the base phase and the other having the characteristics averaged over the body volume.