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Heat Transfer Research
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Heat Transfer Research

DOI: 10.1615/HeatTransRes.v36.i8.30
pages 641-653

Application of the "Geometrical-Optical" Asymptotic Method for Accounting the Impacts of a Complex-Shape Boundary of the Random Region on Multidimensional Nonlinear Irregular Thermal Fields

G. A. Nesenenko
N. E. Bauman Moscow State Technical University, Moscow, Russia


A method for obtaining approximate analytical solutions of nonlinear boundary-value problems, formulated for multidimensional parabolic equations with a small parameter ε > 0 and the Laplace operator has been proposed and substantiated. Such problems are called singularly perturbed boundary-value problems or irregular boundary-value problems. Regions, in which solutions of the above irregular heat-conduction problems are sought by the proposed method, can have a random shape, and nonlinear boundary conditions can be specified at the boundaries. The approximate solutions are represented by the Poincare asymptotics, containing both powers of a small parameter ε > 0 and powers of respective boundary layer variables. The Poincare asymptotic coefficients depend on the geometrical characteristics of the surface, bounding the region in which the solution is analyzed; in so doing, they do not depend on the small parameter ε > 0. To calculate explicitly the coefficients of asymptotic expansion, a mathematically correct analysis of integral solution representations, written by means of respective Green functions, is used.

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