The analytical solution of the 3D model with Robin's boundary conditions for 2 peat layers

Abstract

<p>In this paper we consider averaging methods for solving the 3-D boundary value problem in domain containing 2 layers of the peat block. We consider the metal concentration in the peat blocks. Using experimental data the mathematical model for calculation of concentration of metal in different points in every peat layer is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations of second order with piece-wise diffusion coefficients in every direction and peat layers.</p><p>The special parabolic and exponential spline, which interpolation middle integral values of piece-wise smooth function, are considered. With the help of this splines is reduce the problems of mathematical physics in 3-D with piece-wise coefficients to respect one coordinate to problems for system of equations in 2-D. This procedure allows reduce the 3-D problem to a problem of 2-D and 1-D problems and the solution of the approximated problem is obtained analytically.</p><p>The solution of corresponding averaged 2-D initial-boundary value problem is obtained also numerically, using for approach differential equations the discretization in space applying the central differences. The approximation of the 2-D non-stationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.</p>

On a 3d Initial-Boundary Value Problem for Determining the Dynamics of Impurities Concentration in a Horizontal Layered Fine-Pore Medium

“…do not depend on time , t then the model (1)(8) will completely coincide with the mathematical model (1.1) from [1], in which the physical interpretations of all the initial data -numerical parameters and functions are exhaustively described. Therefore, in this paper we will not describe the physical meaning of the initial numerical parameters and functions of the model (1)(8): they have the same meaning as in [1]. End of Remark 1.…”

“…in width and in length), and, then, the obtained 2D problems have been solved by the standard/classical analytical methods. As opposed to the combinational approach suggested in [1], in the present paper, we do not use approximation methods at all (basically, result of application of the averaging method always is approximate).…”

“…In the present paper, we propose an analytical approach for solving the 3D unsteady-state boundary-value problem for the second-order parabolic equation with the third type boundary conditions in two-layer rectangular parallelepipedic domain. Such type problems arise in particular at study of metal concentration dynamics in the peat blocks (for instance, see [1]- [5] and respective references given in these). Mathematical statement of the considered problem is taken from the article [1], where the problem was solved by combination of the two approaches: firstly, the averaging method in the vertical direction (i.e.…”

“…Such type problems arise in particular at study of metal concentration dynamics in the peat blocks (for instance, see [1]- [5] and respective references given in these). Mathematical statement of the considered problem is taken from the article [1], where the problem was solved by combination of the two approaches: firstly, the averaging method in the vertical direction (i.e. in height) and two horizontal directions (i.e.…”

On Mathematical Modelling of the 2-D Filtration Problem in Porous Axial Symmetrical Cylinder

“…The task of sufficient accuracy numerical simulation of quickly solution 3-D problems for mathematical physics in multilayered media is important in known areas of the applied sciences -heat transfer in multilayered media, for example, calculation of the concentration of metals in peat layers [16], the heat and moisture transfer processes in the porous multilayered media layer, for example, mathematical modelling of moistening and drying process in the wood-block [1].…”