RÉSUMÉ

A method for obtaining approximate analytical solutions of nonlinear boundary-value problems of irregular heat conduction has been proposed and substantiated. The regions, where solutions can be found by the suggested method, can have a random shape and nonlinear conditions can be specified on their boundaries. The developed "geometrical-optical" asymptotic method allows one to find approximate analytical solutions in the form of asymptotic Poincare expansions whose coefficients are calculated explicitly. The report gives as an example the results of analytical-numerical parametric analysis of irregular (i.e., singularly perturbed) temperature fields in a boundary layer on the side of the rectangle on which nonlinear boundary conditions are specified. Two types of nonlinear boundary conditions are discussed, viz., exponential (Arrhenius) type and Stefan-Boltzmann type. A set of parameters has been found for which the initial Gaussian-type distribution results, in the boundary layer, in a local nonlinear enhancement of the thermal field, i.e., in "thermal resonance".