ANALYTICAL AND APPROXIMATE SOLUTIONS OF THE TWO-DIMENSIONAL LAPLACE EQUATION IN CYLINDRICAL COORDINATES

This paper investigates a steady-state heat conduction problem in a multilayer cylindrical domain with nonhomogeneous boundary conditions. Two approaches for solving the direct problem are considered: an analytical method based on separation of variables and Fourier series expansion, and a numerical method based on the finite difference approximation combined with the Successive Over-Relaxation (SOR) algorithm. An exact analytical solution of the Laplace equation in polar coordinates is derived for a threelayer cylindrical system with continuity conditions at layer interfaces and convective heat exchange at the outer boundary. A finite d ifference sc heme is de veloped to ap proximate the go verning eq uation, wh ile the resulting system of linear algebraic equations is solved iteratively using the SOR method.The accuracy of the numerical approach is evaluated through comparison with the analytical solution and experimental temperature measurements. The obtained results demonstrate good agreement between the analytical and numerical solutions. The mean absolute error between the two approaches is 0.134K, the root mean square error is 0.141K, and the maximum absolute error does not exceed 0.209K. Experimental validation shows that both models correctly reproduce the overall temperature distribution in the multilayer cylindrical system. The mean absolute errors relative to experimental data are 2.509K for the analytical solution and 2.638K for the SOR solution. The performed analysis confirms the good a ccuracy and s tability of the developed finite difference algorithm and demonstrates its applicability for modeling heat transfer processes in multilayer cylindrical structures. The proposed approach can be employed in the thermal analysis of pipelines, geothermal systems, insulation structures, and other engineering applications involving layered media.

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