摘要

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.