Дифференциальные уравнения

Davydova M.A., Rublev G.D.

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.

Iskandarov S.

Askhabov S.N.

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.

Sadovnichii V.A., Sultanaev Y.T., Valeev N.F.

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 ????0, find the closest function ????^ such that the first ???? eigenvalues of the Schrodinger operator with potential ????^ coincided with the given values ????* ∈ R, ???? = 1, ????.

Mamatov A.E.

In this paper, we study the direct scattering problem on the semi-axis for the system of Dirac differential equations in the case of finite density with the boundary condition ????1(0) = ????2(0). The spectrum was studied, the resolvent and spectral expansion in terms of eigenfunctions of the Dirac operator were constructed.

Shamolin M.V.

Tensor invariants (first integrals and differential forms) of homogeneous dynamical systems on tangent bundles to smooth three-dimensional manifolds (systems with three degrees of freedom) are presented in this paper. The connection between the presence of such invariants and the complete set of the first integrals, which are necessary for the integration of geodesic, potential, and dissipative systems, is shown. At the same time, the introduced force fields make the considered systems dissipative with dissipation of different signs and generalize the previously considered ones.

Il’in A.V., Fursov A.S.

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.

Kruglov V.E.

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

Emel’yanov D.P.

We consider a Dirichlet boundary problem for an elliptic type equation with a non-regular degeneracy of noninteger order in a rectangle. The coefficients of the differential operator are supposed to be analytic. We build a formal solution by using the method for spectral separation of the singularities. The solution is series where its non-analytic dependency on ???? near point ???? = 0 is written explicitly. We proof the convergence of the series to the classical solution using the Green’s function method.

Metel'skiy A.V.

Елкин В.И.

Khartovskii V.E.

For linear autonomous differential-difference of a neutral type system, the existence criterion was proved and a method was proposed for designing a controller with feedback on the observed output, providing the closed loop system finite stabilization (solution to the problem of complete 0-controllability) and a finite predetermined spectrum. This makes the closed loop system exponentially stable. The constructiveness of the presented results is illustrated by an example.

Tochilin P.A., Chistyakov I.A.

A nonlinear system of ordinary differential equations with control parameters is considered. Pointwise restrictions are imposed on the possible values of these parameters. It is required to solve the problem of transferring the trajectory of the system from an arbitrary initial position to the smallest possible neighborhood of a given target set at a fixed time interval by selecting the appropriate feedback control. To solve this problem, it is proposed to construct a continuous piecewise cubic function of a special kind. The level sets of this function correspond to internal estimates of the solvability sets of the system. Using this function, it is also possible to construct a feedback control function that solves the target control problem at a fixed time interval. The paper proposes formulas for calculating the values of a piecewise cubic function, examines its properties, and considers an algorithm for searching for parameters defining this function.

Naimov A.N., Bystretsky M.V.

In the paper was investigated the a priori estimate and the existence of periodic solutions of a fixed period for a system of second-order ordinary differential equations with the main quasi-homogeneous non-linearity. It is proved that an a priori estimate of periodic solutions takes place if the corresponding unperturbed system does not have non-zero bounded solutions. Under the conditions of an a priori estimate, using methods for calculating the mapping degree of vector fields, a criterion for the existence of periodic solutions under any perturbation from a given class is formulated and proven. The results obtained differ from earlier results in that the set of zeros of the main non-linear part is not taken into account.

Mironenko V.I., Mironenko V.V.