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Vol 61, No 2 (2025)
ORDINARY DIFFERENTIAL EQUATIONS
ON EXISTENCE OF PERIODIC SOLUTIONS OF AN ORDINARY SECOND-ORDER DIFFERENTIAL EQUATION WITH PARAMETER AND DISCONTINUOUS RIGHT-HAND SIDE WITH VARIOUS BOUNDARY CONDITIONS
Abstract
An ordinary second-order differential equation with positive parameter and discontinuous right-hand side which changes its sign at the point of the jump is considered. Various boundary value problems for it are formulated, including mixed and periodic boundary conditions. Theorems on existence of periodic solutions of the studied boundary value problems are established. The obtained results are illustrated by examples.
Differencial'nye uravneniya. 2025;61(2):147–161
147–161
ASYMPTOTICS OF EIGENVALUES AND EIGENFUNCTIONS OF THE STURM–LIOUVILLE OPERATOR WITH SINGULAR POTENTIAL ON A STAR GRAPH. I
Abstract
Spectral problems on a star-graph consisting of three edges with a Sturm–Liouville operator defined on each of them are investigated. The spectral properties of such operators have been studied, in particular, asymptotic formulas for eigenvalues and eigenfunctions of the operator with Dirichlet boundary conditions at free ends and continuity and Kirchhoff conditions at a common vertex have been obtained. The potential in the Sturm–Liouville problem is assumed to be singular, it is a derivative of a quadratically summable function in sense of distributions.
Differencial'nye uravneniya. 2025;61(2):162-176
162-176
177-189
190-199
ON A PRIORI ESTIMATE OF PERIODIC SOLUTIONS OF THE SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER WITH THE MAIN POSITIVELY HOMOGENEOUS NONLINEARITY
Abstract
For the system of ordinary differential equations of the second order with the main positively homogeneous nonlinearity, an a priori estimate of periodic solutions of a fixed period is investigated. New conditions of a priori estimate are found, in which the influence of the properties of the main nonlinear part, including its set of zeros, is mediated by functional estimates from above and below. The feasibility of the new conditions is investigated for three types of nonlinearities.
Differencial'nye uravneniya. 2025;61(2):200-206
200-206
ON THE SPECTRA OF OSCILLATION EXPONENTS OF A TWO-DIMENSIONAL NONLINEAR SYSTEM AND ITS FIRST APPROXIMATION SYSTEM
Abstract
The sets of values (spectra) of the exponents of oscillation of strict signs, non-strict signs, zeros, roots and hyperroots of solutions of differential systems are studied. Two-dimensional nonlinear systems are constructed, all of whose solutions are infinitely extendable to the right and any of the spectra of their oscillation exponents can coincide with both the segment [0, 1] and with any pre-defined nonempty subset of rational numbers of this segment, while the spectra of linear systems of their first approximation consist of only one element. Moreover, the spectra of the exponents of the original system coincide with the corresponding spectra of the exponents of oscillation of the narrowing of the constructed nonlinear two-dimensional systems to the direct product of any open neighborhood of the zero of the phase plane and the time semi-axis. In addition, the existence of a nonlinear system has been proven, the spectrum of any of the oscillation exponents under consideration of which coincides with an arbitrary predetermined interval of the segment [0, 1], and the corresponding spectra of the system of its first approximation consist of one non-negative number.
Differencial'nye uravneniya. 2025;61(2):207-220
207-220
ON THE ASYMPTOTICS OF SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS OF ODD ORDER
Abstract
The asymptotic behavior for large values of the independent variable of the fundamental system of solutions of linear differential equations generated by a symmetric two-term differential expression of arbitrary odd order is investigated, depending on the coefficients of the highest derivative and the free term.
Differencial'nye uravneniya. 2025;61(2):221-228
221-228
PARTIAL DERIVATIVE EQUATIONS
POISSON FORMULA FOR SOLVING THE RADIAL CAUCHY PROBLEM FOR A SINGULAR ULTRAHYPERBOLIC EQUATION
Abstract
The singular ultrahyperbolic equation (Δ𝐵𝛽)𝑦𝑢= (Δ𝐵𝛾)𝑥𝑢 is considered under the assumption that the I.A. Kipriyanov condition is satisfied, where fractional dimensions of the Δ𝐵𝛾 -operators included in the equation are equal to the same positive number 𝜎. Three types of solutions to the radial Cauchy problem are studied, one of them is based on the T-pseudoshift operator, generalized T-shift and S.A. Tersenov’s method for determining solutions to equations that degenerate on the boundary. Poisson formulas for solving the Cauchy problem for the Euler–Poisson–Darboux equation are given for various values of the parameters in this equation.
Differencial'nye uravneniya. 2025;61(2):229-241
229-241
NUMERICAL METHODS
ON THE ESTIMATION OF THE EXPLICIT EULER METHOD LOCAL ERROR FOR THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS TRANSFORMED TO THE BEST ARGUMENT
Abstract
The paper considers the numerical solution of the Cauchy problem for systems of ordinary differential equations. Special attention is paid to problems with limiting singular points on integral curves. It is known that traditional explicit methods for solving the Cauchy problem are ineffective for this class of problems. Implicit methods are much more difficult to use and do not always lead to the result of the desired accuracy. Therefore, along with traditional methods of numerical integration of the Cauchy problem authors use the method of solution continuation with respect to the best argument (also known as the best parameterization and the arc length method). The best argument is calculated tangentially along the integral curve of the problem under consideration. For the Cauchy problems transformed to the best argument, the authors in this paper present the results of a study of the local error for the numerical solution obtained by the explicit Euler method. An estimate of the numerical solution local error of the numerical solution for the Cauchy problem transformed to the best argument is obtained for the explicit Euler method. Using it, an upper estimate of the local error was obtained and the effectiveness of using the best argument was proved. This is reflected in a decrease of the solution local error for the transformed problem in the neighborhood of the limiting singular points. The theoretical results are compatible with the numerical solution of the ill-conditioned initial value problem of deformable solid mechanics with one limiting singular point.
Differencial'nye uravneniya. 2025;61(2):242-260
242-260
BRIEF MESSAGES
261-267
THE CAUCHY PROBLEM FOR A SYSTEM OF MOMENT THEORY OF ELASTICITY
Abstract
The question of the solvability of the problem of analytical continuation of the solution of the system of equations of the moment theory of elasticity in a region of three-dimensional space based on its values and the values of its stress on a part of the boundary of this region is investigated.
Differencial'nye uravneniya. 2025;61(2):268-274
268-274
CHRONICLE
О СЕМИНАРЕ ПО ПРОБЛЕМАМ НЕЛИНЕЙНОЙ ДИНАМИКИ И УПРАВЛЕНИЯ В МОСКОВСКОМ ГОСУДАРСТВЕННОМ УНИВЕРСИТЕТЕ ИМЕНИ М.В. ЛОМОНОСОВА
Abstract
Ниже публикуются краткие аннотации докладов, состоявшихся в осеннем семестре 2024 г. (предыдущее сообщение о работе семинара дано в журнале “Дифференциальные уравнения”. 2024. Т. 60. № 8; дополнительная информация по адресу iline@cs.msu.ru)
Differencial'nye uravneniya. 2025;61(2):275-288
275-288