Asymptotically Stable Solutions with Boundary and Internal Layers in Direct and Inverse Problems for a Singularly Perturbed Heat Equation with Nonlinear Thermal Diffusion
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引用次数: 0
Abstract
This paper proposes a new approach to the study of direct and inverse problems for a
singularly perturbed heat equation with nonlinear temperature-dependent diffusion, based on the
further development and use of asymptotic analysis methods in the nonlinear singularly perturbed
reaction–diffusion–advection problems. The essence of the approach is presented using the
example 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 for
the existence of classical solutions of the boundary layer type and the type of contrast structures
are formulated, asymptotic approximations of an arbitrary order of accuracy to such solutions are
constructed, algorithms for constructing formal asymptotics are substantiated, and the Lyapunov
asymptotic stability of stationary solutions with boundary and internal layers as solutions of the
corresponding parabolic problems is investigated. A class of nonlinear problems that take into
account 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 in
problems of this type is proved. As applications of this research, methods for solving specific
direct and inverse problems of nonlinear heat transfer related to increasing the operating efficiency
of rectilinear heating elements in the smelting furnaces (heat exchangers) are presented, which
include the calculation of thermal fields in the heating elements and a method for reconstructing
thermal diffusion and heat transfer coefficients based on modeling data.
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