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Vol 60, No 8 (2024)

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ORDINARY DIFFERENTIAL EQUATIONS

MULTIDIMENSIONAL AUTONOMOUS DIFFERENTIAL SYSTEM WITH UNIT MEASURE OF INSTABILITY AND MASSIVE PARTICULAR STABILITY

Bondarev A.A.

Abstract

An example of non-one-dimensional autonomous differential system is constructed, in which, on the one hand, all solutions starting in the exterior of the unit ball, tend to zero with unlimited growth of time, and on the other hand, a relative measure of the initial conditions of those solutions that begin in the ball with a center at the zero and move away from it at a sufficient distance with increasing time, approaches arbitrary close to unity as the radius of the ball tends to zero. The nonlinear system constructed in this work also has a zero linear approximation along the zero solution.
Differencial'nye uravneniya. 2024;60(8):1011-1020
pages 1011-1020 views

BOTTOM ESTIMATES FOR THE MINIMAL EIGENVALUE OF THE BI-LAPLACIAN ON A GRAPH

Kulaev R.C., Karkuzaev S.A.

Abstract

Bottom estimates for the minimum eigenvalues of fourth-order differential operators on graphs are found. An analogue of the Picone identity for a fourth-order equation on a network is established. Comparison theorems of the Sturm type for such an equation are obtained.
Differencial'nye uravneniya. 2024;60(8):1034-1048
pages 1034-1048 views

ANALYTICAL CALCULATION OF FIXED POINT OF OPERATOR GENERATED BY MULTIDIMENSIONAL SYSTEM WITH RELAY HYSTERESIS

Kamachkin A.M., Yevstafyeva V.V., Potapov D.K.

Abstract

We consider a multidimensional system of ordinary differential equations with relay nonlinearity of a hysteresis type in a special form that has made it possible to analytically calculate the fixed point of the operator generated by this system. We propose procedures for selecting a vector defining the location of discontinuity surface (switch surfaces) in the phase space such that there exists a unique fixed point on one of these surfaces. Examples representing the obtained theoretical results are given.
Differencial'nye uravneniya. 2024;60(8):1021-1033
pages 1021-1033 views

PARTIAL DERIVATIVE EQUATIONS

ON A NON-LOCAL PROBLEM FOR THE GELLERSTEDT EQUATION WITH SINGULAR COEFFICIENTS

Mirsaburov M., Turaev R.N.

Abstract

The question of the unambiguous solvability of a non-local boundary value problem with conditions of the Bitsadze–Samarskii and Frankl type for a mixed type equation with singular coefficients is investigated.
Differencial'nye uravneniya. 2024;60(8):1086-1099
pages 1086-1099 views

INITIAL BOUNDARY VALUE PROBLEM FOR THE NONLINEAR MODIFIED BOUSSINESQ EQUATION

Zamyshlyaeva A.A., Bychkov E.V.

Abstract

The problem in Sobolev spaces is investigated for a modified Boussinesq equation with a homogeneous Neumann boundary condition and with classical initial conditions. Based on the compactness method, it is shown that the approximate analytical solution, constructed in the form of Galerkin’s sum over the system of eigenfunctions of the Neumann boundary value problem, *-weakly converges to the exact solution.
Differencial'nye uravneniya. 2024;60(8):1076-1085
pages 1076-1085 views

PHRAGMEN–LINDELOF TYPE THEOREMS

Juraeva U.Y.

Abstract

In this paper the Fragmen-Lindelof type theorems for biharmonic functions are proved.
Differencial'nye uravneniya. 2024;60(8):1063-1075
pages 1063-1075 views

ON THE DETERMINATION OF THE STATIONARY TEMPERATURE IN AN UNLIMITED STRIP

Alimov S.A., Kudaybergenov A.K.

Abstract

The problem of determining the stationary temperature at the upper boundary of the strip under known conditions at the lower boundary is considered. The existence and uniqueness of the solution to this problem are proved.
Differencial'nye uravneniya. 2024;60(8):1049-1062
pages 1049-1062 views

NUMERICAL METHODS

ASYMPTOTIC PROPERTIES OF PARAMETRIC EIGENVALUE PROBLEMS IN THE HILBERT SPACE

Samsonov A.A.

Abstract

The parametric eigenvalue problem in infinite-dimensional Hilbert space arising in the mechanics of loaded thin-walled structures is investigated. Asymptotic properties of solutions depending on loading parameters are established. The initial infinite-dimensional problem is approximated in a finitedimensional subspace. Theoretical error estimates of approximate solutions are obtained. Effective numerical methods for calculating the main resonance frequency and the corresponding resonance form of vibrations based on asymptotic formulas are proposed.
Differencial'nye uravneniya. 2024;60(8):1112-1123
pages 1112-1123 views

NUMERICAL METHOD FOR SOLVING OF THE DIFFRACTION PROBLEM DESCRIBED BY MAXWELL’S EQUATIONS WITH MESOSCOPIC BOUNDARY CONDITIONS

Eremin Y.A., Lopushenko V.V.

Abstract

A numerical method for solving the diffraction boundary problem for the system of Maxwell’s equations with mesoscopic boundary conditions has been developed and implemented. It is based on the discrete source method. A numerical analysis of the influence of surface quantum effects on the optical characteristics of plasmonic nanoparticles is carried out. It has been established that surface effects have a significant impact on the field characteristics, and the results differ significantly from the case of volumetric effects.
Differencial'nye uravneniya. 2024;60(8):1100-1111
pages 1100-1111 views

BRIEF MESSAGES

WHOLE SOLUTIONS OF ONE CLASS OF ALGEBRAIC DIFFERENTIAL EQUATIONS GENERALIZING BRIOT–BOUQUET TYPE EQUATIONS

Yanchenko A.Y.

Abstract

The paper examines entire solutions of differential equations of Brio–Bouquet type. It is shown that (under some conditions that the polynomials
Differencial'nye uravneniya. 2024;60(8):1131-1136
pages 1131-1136 views

BOUNDARY VALUE PROBLEM FOR ONE DIFFERENTIAL EQUATION WITH VARIABLE COEFFICIENTS AND A FRACTIONAL LIOUVILLE DERIVATIVE

Akhmetshin A.D.

Abstract

A boundary value problem for an equation with variable coefficients containing a fractional Liouville derivative for one of the variables with a beginning at minus infinity is considered. A representation of the solution of the problem and sufficient conditions for its existence and uniqueness have been obtained.
Differencial'nye uravneniya. 2024;60(8):1124-1130
pages 1124-1130 views

CHRONICLE

O SEMINARE PO PROBLEMAM NELINEYNOY DINAMIKI I UPRAVLENIYa PRI MOSKOVSKOM GOSUDARSTVENNOM UNIVERSITETE IMENI M.V. LOMONOSOVA

Il'in A.V.

Abstract

Ниже публикуются краткие аннотации докладов, состоявшихся в весеннем семестре 2024 г. (предыдущее сообщение о работе семинара дано в журнале “Дифференц. уравнения”. 2024. Т. 60. № 2; за дополнительной информацией обращаться по адресу: iline@cs.msu.ru)
Differencial'nye uravneniya. 2024;60(8):1137-1152
pages 1137-1152 views