Asymptotically Stable Solutions with Boundary and Internal Layers in Direct and Inverse Problems for a Singularly Perturbed Heat Equation with Nonlinear Thermal Diffusion

This paper proposes a new approach to the study of direct and inverse problems for asingularly perturbed heat equation with nonlinear temperature-dependent diffusion, based on thefurther development and use of asymptotic analysis methods in the nonlinear singularly perturbedreaction–diffusion–advection problems. The essence of the approach is presented using theexample of one class of one-dimensional stationary problems with nonlinear boundary conditions,for which the case of applicability of asymptotic analysis is singled out. Sufficient conditions forthe existence of classical solutions of the boundary layer type and the type of contrast structuresare formulated, asymptotic approximations of an arbitrary order of accuracy to such solutions areconstructed, algorithms for constructing formal asymptotics are substantiated, and the Lyapunovasymptotic stability of stationary solutions with boundary and internal layers as solutions of thecorresponding parabolic problems is investigated. A class of nonlinear problems that take intoaccount lateral heat exchange with the environment according to Newton’s law is considered.A theorem on the existence and uniqueness of a classical solution with boundary layers inproblems of this type is proved. As applications of this research, methods for solving specificdirect and inverse problems of nonlinear heat transfer related to increasing the operating efficiencyof rectilinear heating elements in the smelting furnaces (heat exchangers) are presented, whichinclude the calculation of thermal fields in the heating elements and a method for reconstructingthermal diffusion and heat transfer coefficients based on modeling data.