The analytical solution of the 3D model with Robin's boundary conditions for 2 peat layers
DOI:
https://doi.org/10.17770/etr2015vol3.618Keywords:
diffusion problem, special splines, analytical and numerical solutionAbstract
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.
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.
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.
Downloads
References
A. Buikis. The analysis of schemes for the modelling same processes of filtration in the underground. Riga, Acta Universitatis Latviensis, 592 1994, pp. 25-32 (in Latvian).
N. S. Bahvalov, N. P. Zhitkov, G. M. Kobelhkov. Numerical mthods. M:, Nauka, 1987 (in Russian).
A. Buikis. The approximation with splines for problems in layered systems. Riga, Acta Universitatis Latviensis, 592 1994, pp. 135-138 (in Latvian).
I. Kangro, H. Kalis, A. Gedroics, E. Teirumnieka, E. Teirumnieks. On mathematical
modllingof metals distributions in peat layers. Mathematical Modelling and Analysis, 19:4 2014, pp. 568-588. Taylor & Francis, Mortimer House, 37-41, London W1T 3JH, UK.
E. Teirumnieka, E. Teirumnieks, I. Kangro, H. Kalis, A. Gedroics. The mathematical modeling of Ca and Fe distribution in peat layers. Proceed. of the 8-th int. scientific practical conference "Environment. Technology. Resources.", pp. 40-47, Rezekne higher education institution, June 20-22, 2 2011.
C. Henry Edwards, David E. Penny. Differential equations and boundary value problems, computing and modelling. Pearson, Prentice Hal, third edition, 2008 (in Russian).
