Vol 60, No 4 (2024)

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 reactiondiffusion-advection problems. The essence of the approach is presented using the example of a class of one-dimensional stationary problems with nonlinear boundary conditions, for which the case of applicability of asymptotic analysis is highlighted. 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 of 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 to 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 proven. As applications of the study, 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: the calculation of thermal fields in the heating elements and the method for restoring the coefficients of thermal diffusion and heat transfer from modeling data.

A nonlinear eigenvalue problem for a system of three equations with boundary conditions of the first kind, describing the propagation of electromagnetic waves in a plane nonlinear waveguide, is considered. This problem is two-parameter problem with one spectral parameter and a second parameter arising from an additional condition. This condition connects the constants of integration that arise when finding the first integrals of the system. The existence of nonlinearizable solutions to the problem is proven.

For any finite set of non-negative numbers containing zero, a two-dimensional linear homogeneous differential system is constructed (periodic if all elements of a given set are pairwise commensurate), in which the spectra of the oscillation exponents of signs, zeros, roots and hyper roots coincide with this set, and all the values of these indicators are essential.

The article obtains two-sided a priori estimates for the solution of a homogeneous third-order Volterra integro-differential equation with power-law nonlinearity and a difference kernel. It is shown that the lower a priori estimate, which plays the role of a weight function when constructing a metric in the cone of the space of continuous functions, is unimprovable. Using these estimates, using the method of weight metrics (analogous to A. Bielecki’s method), a global theorem on the existence, uniqueness and method of finding a nontrivial solution to the initial problem for the specified integro-differential equation in the class of non-negative continuous functions on the positive half-axis is proved. It is shown that the solution can be found by the method of successive approximations and an estimate of the rate of their convergence to the exact solution is obtained. Examples are given to illustrate the results obtained.

An approach is proposed to constructing a digital controller that stabilizes a continuously switched linear system with commensurate delays in control during slow switchings. The approach to stabilization consistently includes the construction of a switchable continuous-discrete closed-loop system with a digital controller, the transition to its discrete model, represented in the form of a switched system with modes of different orders, simultaneous stabilization of the subsystems of the resulting discrete model and calculation of the delay time that ensures the stability of the original switched system , closed by the found regulator.