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An approximate analytic solution of a particular boundary value problem. (English) Zbl 0992.35022

Summary: This note is concerned with the three-dimensional quasi-steady-state heat conduction equation subject to certain boundary conditions in the whole \(x'y'\)-plane and finite in \(z'\)-direction. This type of boundary value problem arises in laser welding process. The solution to this problem can be represented by an integral using Fourier analysis. This integral is approximated to obtain a simple analytic expression for the temperature distribution.

MSC:

35C05 Solutions to PDEs in closed form
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
44A15 Special integral transforms (Legendre, Hilbert, etc.)