We consider averaging methods for solving the 3-D boundary-value problem of second order in multilayer domain. The special hyperbolic and exponential type splines, with middle integral values of piece-wise smooth function interpolation are considered. With the help of these splines the problems of mathematical physics in 3-D with piece-wise coefficients are reduced with respect to one coordinate to 2-D problems. This procedure also allows to reduce the 2-D problems to 1-D problems and the solution of the approximated problemsa can be obtained analytically. In the case of constant piece-wise coefficients we obtain the exact discrete approximation of a steady-state 1-D boundary-value problem.
The solution of corresponding averaged 3-D initial-boundary value problem is also obtained numerically, using the discretization in space with the central diferences. The approximation of the 3-D nonstationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.
Buikis, A., Kalis, H., & Kangro, I. (2016). Special Splines of Exponential Type for the Solutions of Mass Transfer Problems in Multilayer Domains. Mathematical Modelling and Analysis, 21(4), 450-465. https://doi.org/10.3846/13926292.2016.1182594
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