Differencial'nye uravneniya
The journal publishes articles and reviews, chronicles of scientific life, anniversary articles and obituaries.
The journal is aimed at mathematicians, scientists and engineers who use differential equations in their research, at teachers, graduate students and students of natural science and technical faculties of universities and universities.
The journal is peer-reviewed and is included in the List of the Higher Attestation Commission of Russia for publishing works of applicants for academic degrees, as well as in the RISC system.
The journal was founded in 1965.
ISSN (print): 0374-0641
Media registration certificate: № 0110211 от 08.02.1993
Founder: Department of Informatics, Computer Science and Automation of the Russian Academy of Sciences, Russian Academy of Sciences (RAS)
Editor-in-Chief: Sadovnichii Victor Antonovich, Member of RAS, Doctor Phys.-Math. Sciences, Rector of Lomonosov Moscow State University
Number of issues per year: 12
Indexation: RISC, Higher Attestation Commission list, RISC core, RSCI, White list (1st level)
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Vol 61, No 4 (2025)
- Year: 2025
- Articles: 10
- URL: https://ogarev-online.ru/0374-0641/issue/view/19610
ORDINARY DIFFERENTIAL EQUATIONS
EXPLICIT FORMULAS FOR COEFFICIENTS IN THE LAPPO-DANILEVSKY SOLUTION OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS
Abstract
In the works of I.A. Lappo-Danilevsky, in particular, the solutions of a system of linear ordinary differential equations in the vicinity of an isolated pole of arbitrary finite order were investigated. For the fundamental matrix of solutions of such a system, a series was obtained that absolutely converges in the punctured (annular) neighborhood of the pole. At the same time, recurrent relations of a rather complex type were found for the numerical coefficients of the specified series, which do not depend on the type of the system of equations. In this paper, explicit formulas for these coefficients are obtained for the first time. As an example, the results obtained are used to find the analytical formula for the trace of the monodromy matrix of an arbitrary regular singular point of the specified system of equations.
Differencial'nye uravneniya. 2025;61(4):435-447
435-447
EXISTENCE THEOREM FOR WEAK SOLUTIONS FOR STOCHASTIC DIFFERENTIAL-DIFFERENCE HYBRID SYSTEM WITH CONTINUOUS COEFFICIENTS
Abstract
The stochastic differential-difference hybrid system with two types of equation is examined: first, a differential equation with feedback, and second, a difference equation with delay. The existence theorem for weak solutions for this system with continuous coefficients is proved.
Differencial'nye uravneniya. 2025;61(4):448-460
448-460
DISTRIBUTION OF THE SPECTRUM OF THE WEBER OPERATOR PERTURBED BY THE DIRAC DELTA FUNCTION
Abstract
In a Hilbert space 𝐿2[0,+∞) the Sturm–Liouville operator generated by a differential expression of a special type containing the Dirac delta function with zero boundary condition is investigated. We prove that the eigenvalues 𝜆𝑛 of this operator satisfy the certain inequalities. The problem on the location of the first eigenvalue 𝜆1 depending on the parameters of the differential expression is solved. In particular, we obtain conditions under which 𝜆1 is negative.
Differencial'nye uravneniya. 2025;61(4):461-471
461-471
PARTIAL DERIVATIVE EQUATIONS
CORRECTNESS ANALYSIS OF THE BOUNDARY VALUE PROBLEM FOR STATIONARY MAGNETOHYDRODYNAMICS EQUATIONS WITH VARIABLE COEFFICIENTS
Abstract
The boundary value problem for stationary equations of magnetohydrodynamics of viscous heatconducting fluid with variable coefficients considered under the Dirichlet condition for velocity and mixed boundary conditions for electromagnetic field and temperature is studied. Sufficient conditions on the initial data, providing global solvability of the mentioned problem and local stability of its solution, are established.
Differencial'nye uravneniya. 2025;61(4):472-489
472-489
BOUNDARY VALUE PROBLEM FOR STATIONARY MAGNETOHYDRODYNAMICS EQUATIONS OF HEAT-CONDUCTING FLUID WITH VARIABLE COEFFICIENTS
Abstract
The global solvability and local uniqueness of boundary value problem’s solutions for stationary magnetic hydrodynamic equations for heat conducting fluid with variable coefficients are proved. Maximum and minimum principle for the temperature is established.
Differencial'nye uravneniya. 2025;61(4):490-503
490-503
CLASSICAL SOLUTION OF A MIXED PROBLEM WITH DIRICHLET AND WENTZEL CONDITIONS FOR THE BIWAVE EQUATION WITH NONLINEAR LOWER ORDER TERMS
Abstract
For the hyperbolic biwave equation with nonlinear lower terms given in the first quadrant of Euclidean space, we consider a mixed problem in which the Cauchy conditions are specified on the spatial semi-axis, and Dirichlet and Wentzel conditions are specified on the temporal semi-axis. The solution is constructed by the method of characteristics in an implicit form as a solution of some integrodifferential equations. The solvability of these equations, as well as the dependence on the initial data and the smoothness of their solutions, is studied using the parameter continuation method and a priori estimates. For the problem under consideration, the uniqueness of the solution is proved and conditions under which there exists a classical solution are established. If the matching conditions are not met, then a problem with conjugation conditions is constructed, and if the data is not smooth enough, then a mild solution is constructed.
Differencial'nye uravneniya. 2025;61(4):504-522
504-522
ELASTIC JUNCTIONS OF A PLATE WITH RODS AND SELF-ADJOINT EXTENSIONS OF DIFFERENTIAL OPERATORS
Abstract
We construct asymptotics of natural oscillations of elastic junctions composed of a thin horizontal plate and several vertical rods joined to it. This construction is rigidly fixed along the plate edge and the exterior end-faces of the rods while physical properties of its elements are chosen such that in the mid-frequency range of the spectrum the limiting spectral problems consists of a self-adjoint operator obtained as extensions of differential operators, namely, a bi-harmonic one in plate’s longitudinal section and ordinary second-order differential operators at rod’s axes. The low-frequency range of the spectrum is formed by eigenvalues of the Dirichlet problem for ordinary forth-order differential operators describing transverse oscillations of rods with fixed ends. Justification of asymptotic formulas is performed by means of anisotropic Korn’s inequality and the classical lemma on “almost eigenvalues”.
Differencial'nye uravneniya. 2025;61(4):523-544
523-544
OPERATOR METHODS FOR INVESTIGATING STABILITY AND BIFURCATION PROBLEMS IN A SYSTEM REACTION–DIFFUSION AND THEIR APPLICATIONS
Abstract
The article discusses the issues of studying stability and bifurcations in the reaction–diffusion system in a bounded domain with homogeneous Neumann boundary conditions. The main results of the article concern the study of problems of local bifurcations in the vicinity of spatially homogeneous equilibrium positions. A general scheme is proposed that allows us to obtain new formulas for studying the main characteristics of multiple equilibrium bifurcation and bifurcation Andronov–Hopf: sufficient signs of bifurcations, their type, approximate construction of solutions, stability analysis. The proposed approaches do not require complex and cumbersome transformations, the results obtained are brought to calculation formulas and algorithms. Some applications in problems of diffusion instability and corresponding bifurcations in reaction–diffusion systems are also discussed. The distributed model of the “brusselator” is considered as the main illustrative example.
Differencial'nye uravneniya. 2025;61(4):545-562
545-562
INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS
563-569
FINITE VELOCITY OF PROPAGATION OF PERTURBATIONS FOR A ONE-DIMENSIONAL WAVE INTEGRO-DIFFERENTIAL EQUATION WITH A FRACTIONAL-EXPONENTIAL MEMORY FUNCTION
Abstract
The paper studies a Volterra integro-differential equation, the main part of which is a one-dimensional wave equation perturbed by an integral operator of the Volterra convolution type (wave equation with memory). The kernel function of the integral operator is a sum of fractional exponential functions (Rabotnov functions) with positive coefficients. The issue of the influence of the integral operator on the velocity of propagation of disturbances in the initial value problem for the wave equation with memory is studied. The Volterra integro-differential equation under study describes oscillations of a one-dimensional viscoelastic rod, as well as the process of heat propagation in media with memory (Gurtin–Pipkin equation).
Differencial'nye uravneniya. 2025;61(4):570-576
570-576