Hyperbolic type splines for solving stationary diffusion problems in 3-d domain with special source functions
2025
Kalis, Harijs | Kangro, Ilmars | Aboltins, Aivars
In the study of various spatial engineering problems (e.g., heat and mass transfer in multilayer media, diffusion and combustion processes), it is necessary to use 3-D partial differential equations (PDE), the solving of which is difficult. Therefore, in solving these problems, we apply the conservative averaging method. The conservative averaging method as an approximate analytical and numerical method for solving PDE or their systems with piece-wise constant (continuous) coefficients is under question. We consider averaging methods for solving the stationary 3-D boundary value problem of second order with piece-wise parameters in the 3-D domain for special source function. The hyperbolic-type splines, which interpolate middle integral values of a piece-wise smooth function, are considered. With the help of these splines, some boundary value problems of mathematical physics in 3-D with piece-wise coefficients are reduced to boundary value problems for ordinal differential equations in 1-D for one coordinate. The usage of this spline allows for diminishing the dimensions of the initial problem per one. The spline solution is used for different coordinates, in Cartesian coordinates, in cylindrical coordinates with axial symmetry and in spherical coordinates with axial symmetry. The analytical solution of the 1-D problem (boundary value problem for the ordinal differential equation) was compared with the corresponding spline function solution in the previously mentioned coordinates. Calculations to test theoretical assumptions and perform numerical experiments were proceeded with MATLAB.
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出版者 Latvia University of Life Sciences and Technologies