


Vol 60, No 8 (2024)
ORDINARY DIFFERENTIAL EQUATIONS
MULTIDIMENSIONAL AUTONOMOUS DIFFERENTIAL SYSTEM WITH UNIT MEASURE OF INSTABILITY AND MASSIVE PARTICULAR STABILITY
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



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



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



PARTIAL DERIVATIVE EQUATIONS



INITIAL BOUNDARY VALUE PROBLEM FOR THE NONLINEAR MODIFIED BOUSSINESQ EQUATION
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









NUMERICAL METHODS
ASYMPTOTIC PROPERTIES OF PARAMETRIC EIGENVALUE PROBLEMS IN THE HILBERT SPACE
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



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



BRIEF MESSAGES



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



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


