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Vol 61, No 5 (2025)
ЛЮДИ НАУКИ
EVGENIY ALEKSANDROVICh BARABANOV (11.11.1958–03.04.2025)
Differencial'nye uravneniya. 2025;61(5):579–580
579–580
ORDINARY DIFFERENTIAL EQUATIONS
SOME VARIATIONAL PRINCIPLES FOR SELF-ADJOINT HAMILTONIAN SYSTEMS
Abstract
In the paper a problem on the connection of the self-adjoint Hamiltonian systems theory with the operators in Banach spaces triplets, and on achievement of the estimates of negative eigenvalues of self-adjoint Hamiltonian systems using this connection is considered.
Differencial'nye uravneniya. 2025;61(5):581–595
581–595
ON VARIOUS RADIAL PROPERTIES OF A DIFFERENTIAL SYSTEM
Abstract
Various qualitative properties of a differential system related to the behavior of its solutions starting near zero are considered: stability and asymptotic stability, complete oscillation, wandering, and rotation, as well as complete negations of each of these properties. Logical connections of their various radial and general radial varieties are studied both with each other and with the corresponding complete properties, as well as with the measures of these properties.
Differencial'nye uravneniya. 2025;61(5):596–605
596–605
PARTIAL DERIVATIVE EQUATIONS
MODEL FIRST BOUNDARY VALUE PROBLEM FOR PARABOLIC SYSTEM IN ZYGMUND SPACES
Abstract
The first boundary value problem for the Petrovsky uniformly parabolic system of the second order with one spatial variable is considered. The coefficients of the system are assumed to be constant, and the domain is a half-strip.The solvability of the problem in the scale of anisotropic Zygmund spaces is established.
Differencial'nye uravneniya. 2025;61(5):606–617
606–617
CLASSICAL SOLUTION OF THE FIRST MIXED PROBLEM FOR THE WAVE EQUATION IN CYLINDRICAL DOMAIN IN ODD DIMENSIONAL SPACE
Abstract
The classical solution of the first mixed problem for the wave equation in the cylindrical area in the space with odd number of dimensions is considered. The solution is constructed using the spherical averages operators which allows the reduction of the equation to the first mixed problem for the one-dimensional wave equation. Using the characteristics method the explicit solution is obtained and necessary and sufficient matching conditions for the existance of the unique classical solution are proved.
Differencial'nye uravneniya. 2025;61(5):618–627
618–627
AN INVERSE PROBLEM FOR ELECTRODYNAMIC EQUATIONS WITH A NONLINEAR CURRENT DEPENDENCE OF A TENSION
Abstract
The system of Maxwell equations in which a current depends nonlinearly of the electrical tension is considered. In the studying case, it is determined of 4 coefficients depended of space variables. These coefficients are supposed to be finite functions with a support located within ball 𝐵(𝑅) of radius 𝑅. For electrodynamic equations a problem of falling down of a plane running wave with a strong front on the inhomogeneity localized in ball 𝐵(𝑅) is posed. A formula for calculation of an amplitude of this wave is derived. In the sequel, an inverse problem of finding 4 coefficients whose determine the current is considered. For this goal the amplitudes formula for different directions of falling waves is used for points at a part of the boundary of 𝐵(𝑅). It is demonstrated that this inverse problem is decomposed at 4 separated problems. One of them is the usual X-ray tomography problem, when the remain 3 others problems are identical problems of the integral geometry for a family of strait lines. In the latter problems, integrals of an unknown function is given along strait lines with a weight function which depends on the finding coefficients after solving the tomography problem. Arising problems of the integral geometry is studied and stability estimate of its solutions is found.
Differencial'nye uravneniya. 2025;61(5):628–639
628–639
INTEGRAL EQUATIONS
MATHEMATICAL MODELING OF SCALE-STRUCTURAL FAILURE AT PROGRAM CYCLIC LOADING OF METALS AND ALLOYS
Abstract
Relations for the failure probability at the micro-, meso-, and macrolevels and fatigue curves on defect levels at the program loading are proposed. The results of calculations for 0,25 % carbon steel at the loading of two or three blocks with different amplitudes and cycle numbers, steel 45 with different distributions of stress amplitudes and titanium alloy TC21 at symmetric loading, each block of which consists of two amplitudes of different numbers of cycles are discussed. For the materials considered, the model describes the evolution of brittle failure and the fatigue curve on fracture at symmetrical loading well. The scope of applicability of the model for program loadings is determined. It well describes the fatigue curve on fracture in a range of 𝑁𝑓 ⩾ 106 number of cycles and program loadings, in which the maximum stress values on average do not exceed the endurance by more than 30 %.
Differencial'nye uravneniya. 2025;61(5):640–658
640–658
CONTROL THEORY
MAXIMUM PRINCIPLE IN THE LINEAR MINIMUM-TIME CONTROL PROBLEM
Abstract
In a minimum-time control problem, both with an autonomous and with a non-autonomous system, the question of the sufficient conditions for the maximum principle is solved. The question of the uniqueness of optimal control is also considered. New conditions are obtained that guarantee the sufficiency of the maximum principle in terms of the geometry of the reachability set and the geometry of the control set. Examples that demonstrate the non-improvability of the obtained results are considered.
Differencial'nye uravneniya. 2025;61(5):659–674
659–674
A SUFFICIENT CONDITION FOR THE CONSISTENCY OF A PIECEWISE LINEAR APPROXIMATION OF A NOLINEAR AFFINE SYSTEM
Abstract
For a controlled nonlinear affine system, a piecewise linear approximation in the form of a switched affine system is considered. The concept of consistency of a piecewise linear approximation is introduced when the system is closed by a variable structure controller. The consistency of the approximation ensures the equality of the graphs of discrete states of the nonlinear system itself and its piecewise linear approximation. A sufficient condition for the consistency of the approximation is obtained and an approach to numerical verification of its fulfillment is proposed.
Differencial'nye uravneniya. 2025;61(5):675–684
675–684
NUMERICAL METHODS
NUMERICAL SOLUTION OF INTEGRAL EQUATIONS OF THE THIRD KIND WITH FIXED SINGULARITIES OF THE KERNEL
Abstract
A linear integral equation of the third kind with fixed singularities in the kernel is studied. For its approximate solution in the space of generalized functions, a special generalized spline method is proposed and substantiated. Optimality of the method in order of accuracy is proved.
Differencial'nye uravneniya. 2025;61(5):685–696
685–696
A POSTERIORI IDENTITIES FOR THE GENERALISED STOKES PROBLEM
Abstract
In the paper, functional identities for the difference between a given function and exact solution of the generalized Stokes problem are derived. The restrictions on the form of such a function are minimal. Actually, they are reduced to the requirement that it must belong to the same functional class as the solution of the problem. The left part of the identity represents a weighted sum of norms and characterizes deviations from the exact velocity and stresses fields. The right part includes a number of summands. Some of them can be directly computed from the problem data and known approximate solutions. Other terms contain unknown functions but can be efficiently estimated. Hence the identity is a basis for getting fully computable two-sided bounds of the distance to the solution of the problem. The identities and the estimates derived from them can be used to estimate errors of approximations generated by various numerical methods. They are true both for solenoidal approximations, as well as for those that satisfy the incompressibility condition only with a certain degree of accuracy. In addition, identities and estimates make it possible to compare exact solutions to problems with different data. Therefore, they open a way to evaluate errors of mathematical models, e.g., those that arise after changing (simplification) of the differential equation or due to replacing the incompressibility condition by weaker conditions.
Differencial'nye uravneniya. 2025;61(5):697–720
697–720