Том 60, № 4 (2024)

Differential equations. 2024;60(4):435-438

435-438

ASYMPTOTICALLY STABLE SOLUTIONS WITH BOUNDARY AND INTERNAL LAYERS IN DIRECT AND INVERSE PROBLEMS FOR THE SINGULARLY PERTURBED HEAT EQUATION WITH A NONLINEAR THERMAL DIFFUSION

Davydova M., Rublev G.

Аннотация

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.

Differential equations. 2024;60(4):439-462

439-462

CHRISTOFFEL–DARBOUX FORMULA FOR POLYNOMIAL EIGENFUNCTIONS OF SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS

Kruglov V.

Аннотация

Using recurrence relations between any three consecutive polynomial eigenfunctions of second-order linear differential equations, the Christoffel–Darboux formulae for the system of polynomial eigenfunctions of these equations are derived

Differential equations. 2024;60(4):463-471

463-471

ON THE EXISTENCE OF NONLINEARIZABLE SOLUTIONS IN A NONCLASSICAL TWO-PARAMETER NONLINEAR BOUNDARY VALUE PROBLEM

Martynova V.

Аннотация

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.

Differential equations. 2024;60(4):472-491

472-491

OPTIMIZATION INVERSE SPECTRAL PROBLEM FOR THE ONE-DIMENSIONAL SCHRODINGER OPERATOR ON THE ENTIRE AXIS

Sadovnichii V., Sultanaev Y., Valeev N.

Аннотация

We investigate the statement of the optimization inverse spectral problem with incomplete spectral data for the one-dimensional Schr¨odinger operator on the entire axis: for a given potential q₀, find the closest function $\hat{q}$ such that the first m eigenvalues of the Schrodinger operator with potential $\hat{q}$ coincided with the given values $λ_{k}^{*} \in ℝ, k = 1, m$ .

Differential equations. 2024;60(4):492-499

492-499

ON THE REALIZATION OF FINITE ESSENTIAL SPECTRA OF OSCILLATION EXPONENTS OF TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS

Stash A., Loboda N.

Аннотация

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.

Differential equations. 2024;60(4):500-507

500-507

AN INVERSE PROBLEM FOR THE WAVE EQUATION WITH TWO NONLINEAR TERMS

Romanov V.

Аннотация

An inverse problem for a hyperbolic equation of the second order containing two nonlinear terms is studied. It consists in recovering coefficients under nonlinearities. The Cauchy problem with a point source located at point y is considered. This point is a parameter of the problem and runs an spherical surface ???? successively. It is supposed that unknown coefficients are differed from zero in domain be situated inside of ???? only. The trace of a solution of the Cauchy problem is given on ???? for all values of y and for all times closed to moments of arriving of the wave from y to points of ????. It is proved that this information allows to reduce the inverse problem to two problems of the integral geometry solving successively. For latter problems stability estimates are stated.

Differential equations. 2024;60(4):508-520

508-520

INITIAL PROBLEM FOR A THIRD ORDER NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF CONVOLUTION TYPE

Askhabov S.

Аннотация

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.

Differential equations. 2024;60(4):521-532

521-532

CORRECT SOLVABILITY OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS ARISING IN VISCOELASTICITY THEORY

Georgievskii D., Rautian N.

Аннотация

We discuss the issues of correct solvability and exponential stability of solutions of abstract integrodifferential equations with kernels of integral operators of general type from the space of functions integrable on the positive semiaxis. The abstract integro-differential equations are studied in this paper are operator models of viscoelasticity theory problems. The proposed approach to the study of these integro-differential equations is related to the application of the semigroups theory and can also be used to study other integro-differential equations containing integral terms of Volterra convolution type.

Differential equations. 2024;60(4):533-549

533-549

STABILIZATION OF THE SWITCHED SYSTEM WITH COMPARABLE DELAYS DURING SLOW SWITCHINGS

Il’in A., Fursov A.

Аннотация

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.

Differential equations. 2024;60(4):550-560

550-560

SINGULARLY PERTURBED OPTIMAL TRACKING PROBLEM

Sobolev V.

Аннотация

We consider a singularly perturbed optimal tracking problem with a given etalon trajectory in the case of incomplete information about the state vector in the presence of external disturbances. To analyze the differential equations that arise when solving this problem, the decomposition method is used, which is based on the technique of integral manifolds of fast and slow motions.

Differential equations. 2024;60(4):561-576

561-576