Contemporary Mathematics. Fundamental Directions
type: channel
description: Editor-in-Chief: Alexander L. Skubachevskii, Professor, Doctor of Physical and Mathematical Sciences ISSN: 2413-3639 (print), 2949-0618 (online). Founded in 2003 г. Publication frequency: quarterly Open Access: . APC: no article processing charge Peer-Review: blind Publication language: Rus, Eng Indexation: White List, elibrary.ru, mathnet.ru, Google Scholar, DOAJ, Lens, Research4Life Publisher: Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)
publisher: Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)
language: ru-RU
publication name: Contemporary Mathematics. Fundamental Directions
issn: 2413-3639
copyright: The right of authorship belongs to the authors of articles. The authors transfer the rights to use the article (including the use and distribution of an article in the Open Access) to the publisher of the journal on a non-exclusive license (Publishing Agreement (Public Offer)). At the same time, authors remain full rightsholders. Publishing Agreement (Public Offer) to Publish an Article in an Academic Periodical "Contemporary Mathematics. Fundamental Directions" The Federal State Autonomous Educational Institution of Higher Education "Peoples’ Friendship University of Russia named after Patrice Lumumba” (RUDN University), represented by the Vice-Rector for Scientific Work Andrey Aleksandrovich Kostin, acting under a Power of Attorney No. 0036-03/21-163 dated 12.10.2021, hereinafter referred to as the Publisher, on the one hand, hereby offers to the public at large, hereinafter referred to as the Author, on the other hand, hereinafter collectively referred to as the Parties, to enter into this agreement, hereinafter referred to as the Agreement, regarding publication of scholarly materials, hereinafter referred to as the Article, in the Journal named 'Contemporary Mathematics. Fundamental Directions', hereinafter referred to as the Journal, under the following terms. 1. General Terms and Conditions 1.1. Pursuant to Clause 2 of the Article 437 of the Russian Federation Civil Code, this Agreement shall be recognized as the public offer, hereinafter referred to as the Offer. Subject to Article 438 of the Russian Federation Civil Code, this Offer shall be deemed to have been completely and irrevocably accepted after the Author had submitted his/her materials by uploading those onto the network electronic system of the article acquisition for review available at the respective section of the Journal's site at URL: http://journals.rudn.ru/index.php/CMFD/author/submit/1 on the Information and Communication Network known as the Internet, hereinafter referred to as the Intern…
items: On one method for solving the initial-boundary value problem for the Gardner equation
items: Modeling of evolutionary strategies of interacting populations in a heterogeneous habitat
items: Modeling of the study of viscoelastic deformation of elastic bodies
items: Existence and uniqueness of the solution of the initialboundary value problem for one-dimensional equations of the dynamics of a compressible viscous mixture
items: Lefschetz formula for nonlocal elliptic problems associated with fibration
items: Continuous generation population model with discontinuous life cycle characteristics
items: Equiconvergence of expansions in root functions of a differential operator and in a trigonometric Fourier series
items: The second four-electron singlet in the Hubbard impurity model
items: Oscillations of a viscous fluid with an inertial free surface
items: Spectral decomposition of self-adjoint operators in Pontryagin and Krein spaces

On one method for solving the initial-boundary value problem for the Gardner equation
type: item
description: In this paper, the efficient solution is considered for an initial-boundary value problem for the Gardner equation: a spatially one-dimensional nonlinear evolution equation describing a broad class of dispersive autowave processes. A numerical-analytical method is proposed based on a combination of explicit and implicit time discretization schemes for various terms of the differential operator. A new efficient algorithm is developed to solve a sequence of auxiliary linear problems, relying on analytical representations using an explicit form of the fundamental system of solutions. An example of a numerical solution of the initial-boundary value problem for the Gardner equation is considered, and the result is compared with a known exact solution of the solitary traveling wave type.
creator: S. I. Bezrodnykh
creator: S. V. Pikulin
rights: Copyright (c) 2025 Bezrodnykh S.I., Pikulin S.V. https://creativecommons.org/licenses/by-nc/4.0
license: https://creativecommons.org/licenses/by-nc/4.0
date: 2025-10-15
publication date: 2025-10-15
volume: 71
number: 3
starting page: 353
ending page: 369
doi: 10.22363/2413-3639-2025-71-3-353-369

Modeling of evolutionary strategies of interacting populations in a heterogeneous habitat
type: item
description: Using a predator-prey model in a heterogeneous environment, we have created a mathematical model that describes the interaction between populations with different evolutionary strategies. The model is based on partial differential equations and allows for the consideration of multifactor taxis. We propose modified functions of local predator-prey interactions, which provide a variety of evolutionary strategies for the system. Key parameters responsible for the formation of ideal free distribution strategies have been investigated, and conditions for the parameters of diffusion and migration have been given under which ideal free distribution-like strategies can be implemented. Results from computational experiments demonstrating stationary and oscillating modes have been presented.
creator: P. A. Zelenchuk
creator: V. G. Tsybulin
rights: Copyright (c) 2025 Zelenchuk P.A., Tsybulin V.G. https://creativecommons.org/licenses/by-nc/4.0
license: https://creativecommons.org/licenses/by-nc/4.0
date: 2025-10-15
publication date: 2025-10-15
volume: 71
number: 3
starting page: 370
ending page: 384
doi: 10.22363/2413-3639-2025-71-3-370-384

Modeling of the study of viscoelastic deformation of elastic bodies
type: item
description: This article proposes a numerical-analytical method for solving linear viscoelasticity problems of an anisotropic solid without the need for explicit analytical representations of creep and relaxation kernels. The approximate solution of integral equations is based on the direct use of experimental data, previously smoothed and filled with a finer mesh. Thus, solving boundary-value problems of viscoelasticity is reduced to solving elasticity problems at an arbitrary point in time.
creator: R. N. Neskorodev
creator: A. V. Zyza
rights: Copyright (c) 2025 Neskorodev R.N., Zyza A.V. https://creativecommons.org/licenses/by-nc/4.0
license: https://creativecommons.org/licenses/by-nc/4.0
date: 2025-10-15
publication date: 2025-10-15
volume: 71
number: 3
starting page: 385
ending page: 394
doi: 10.22363/2413-3639-2025-71-3-385-394

Existence and uniqueness of the solution of the initialboundary value problem for one-dimensional equations of the dynamics of a compressible viscous mixture
type: item
description: In this paper, an initial-boundary value problem is studied for one-dimensional equations of the dynamics of a compressible viscous mixture. A theorem is proved for the existence and uniqueness of a solution to the initial-boundary value problem without any restrictions on the structure of the viscosity matrix other than the standard physical requirements of symmetry and positive definiteness.
creator: V. Yu. Nogovishcheva
creator: D. A. Prokudin
rights: Copyright (c) 2025 Nogovishcheva V.Y., Prokudin D.A. https://creativecommons.org/licenses/by-nc/4.0
license: https://creativecommons.org/licenses/by-nc/4.0
date: 2025-10-15
publication date: 2025-10-15
volume: 71
number: 3
starting page: 395
ending page: 416
doi: 10.22363/2413-3639-2025-71-3-395-416

Lefschetz formula for nonlocal elliptic problems associated with fibration
type: item
description: Elliptic operator complexes associated with fibration are considered. The Atiyah-Bott-Lefschetz formula for endomorphisms of such complexes is given. The proof is based on the stationary phase method. Wavefronts of distributions are used to estimate the remainder term.
creator: N. R. Orlova
rights: Copyright (c) 2025 Orlova N.R. https://creativecommons.org/licenses/by-nc/4.0
license: https://creativecommons.org/licenses/by-nc/4.0
date: 2025-10-15
publication date: 2025-10-15
volume: 71
number: 3
starting page: 417
ending page: 442
doi: 10.22363/2413-3639-2025-71-3-417-442

Continuous generation population model with discontinuous life cycle characteristics
type: item
description: Traditionally, continuous models of mathematical biology are focused on the dynamics of interacting populations as stationary homogeneous entities. The state of the populations in the equations is governed by factors common to all individuals \( \forall t,N(t) \): reproductive efficiency, mortality, living space limitations, or resource limitations. Many species exist with nonoverlapping generation sequences, replacing each other under different seasonal conditions. The number of annual generations is an important characteristic of the ecology of a species when occupying a new range. The length of the life cycle and the index of reproductive activity r in adjacent generations of insects in a range vary due to the need for wintering. Fluctuations in these values affect rapid population outbreaks. It is shown that the use of discrete models \( x_{n+1}=\psi(x_n;r)\varphi(x_{n-i})-\Xi \) is unrealistic for fundamental reasons. The appearance of cycles \( p\neq2^i \) in the order of Sharkovsky's theorem is excessive for the analysis of populations and the forecast of mass reproductions of insects. The article proposes a method for organizing models of the conjugate development of a succession of generations in a system of discontinuous differential equations as a sequence of boundary-value problems. The model is event-based redefined to obtain a solution on time intervals corresponding to the conditions of the season. The model taking into account competition and delayed regulation is relevant for the analysis of a sequence of peaks in pest activity, which are characterized by individual extremely numerous generations.
creator: A. Yu. Perevaryukha
rights: Copyright (c) 2025 Perevaryukha A.Y. https://creativecommons.org/licenses/by-nc/4.0
license: https://creativecommons.org/licenses/by-nc/4.0
date: 2025-10-15
publication date: 2025-10-15
volume: 71
number: 3
starting page: 443
ending page: 451
doi: 10.22363/2413-3639-2025-71-3-443-451

Equiconvergence of expansions in root functions of a differential operator and in a trigonometric Fourier series
type: item
description: We consider a non-self-adjoint ordinary differential operator defined on a finite interval by an \( n \)th-order linear differential expression with a nonzero coefficient of the \( (n-1) \)th derivative and two-point Birkhoff regular boundary conditions. We study the uniform equiconvergence of expansions of a given function in a biorthogonal series in eigenfunctions and associated functions (or, briefly, root functions) of this operator and in an ordinary trigonometric Fourier series, as well as an estimate of the difference of the corresponding partial sums (or, briefly, the rate of equiconvergence) under the most general conditions on the expanded function and the coefficient of the \( (n-1) \)th derivative. We obtain estimates for the difference of the expansions in terms of general (integral) moduli of continuity of the expanded function and the coefficient of the \( (n-1) \)th derivative uniform inside the fundamental interval. From these estimates, corresponding estimates are derived in the case where moduli of continuity are bounded from above by slowly varying functions and, in particular, by logarithmic functions. Based on this, sufficient conditions for equiconvergence in the indicated cases are formulated. These results are obtained using the author's previously obtained estimate for the difference between the partial sums of expansions of a given function in a biorthogonal series in eigenfunctions and associated functions of the differential operator under consideration and in a modified trigonometric Fourier series, as well as analogues of the Steinhaus theorem. The modification of the trigonometric Fourier series consisted in applying a very specific bounded operator to the ordinary trigonometric Fourier series expressed through the coefficient of the \( (n-1) \)th derivative and its inverse operator to the expanded function.
creator: V. S. Rykhlov
rights: Copyright (c) 2025 Rykhlov V.S. https://creativecommons.org/licenses/by-nc/4.0
license: https://creativecommons.org/licenses/by-nc/4.0
date: 2025-10-15
publication date: 2025-10-15
volume: 71
number: 3
starting page: 452
ending page: 477
doi: 10.22363/2413-3639-2025-71-3-452-477

The second four-electron singlet in the Hubbard impurity model
type: item
description: We consider the energy operator of four-electron systems in the Hubbard impurity model and investigate the structure of the essential spectrum and discrete spectra for the second singlet state of the system. It is shown that in the one-dimensional and two-dimensional cases the following situations exist for the essential and discrete spectrum: (a). the essential spectrum of the operator of the second singlet state of four electrons in the Hubbard impurity model consists of a union of eight segments, and the discrete spectrum of the operator consists of six eigenvalues; (b). the essential spectrum of the operator consists of a union of sixteen segments, and the discrete spectrum of the operator consists of fourteen eigenvalues; (c). the essential spectrum of the operator consists of a union of thirteen segments, and the discrete spectrum of the operator consists of nine eigenvalues; (d). the essential spectrum of the operator consists of a union of three segments, and the discrete spectrum of the operator consists of three eigenvalues. In the three-dimensional case the following situations arise: (a). the essential spectrum of an operator consists of unions of eight segments, and the discrete spectrum of the operator consists of six eigenvalues, or the essential spectrum of an operator consists of unions of three segments, and the discrete spectrum of the operator consists of three eigenvalues; (b). the essential spectrum of an operator consists of unions of eight segments, and the discrete spectrum of the operator consists of six eigenvalues; (c). the essential spectrum of an operator consists of unions of sixteen segments, and the discrete spectrum of the operator consists of fourteen eigenvalues; (d). the essential spectrum of an operator consists of unions of three segments, and the discrete spectrum of the operator consists of three eigenvalues.
creator: S. M. Tashpulatov
creator: R. T. Parmanova
rights: Copyright (c) 2025 Tashpulatov S.M., Parmanova R.T. https://creativecommons.org/licenses/by-nc/4.0
license: https://creativecommons.org/licenses/by-nc/4.0
date: 2025-10-15
publication date: 2025-10-15
volume: 71
number: 3
starting page: 478
ending page: 507
doi: 10.22363/2413-3639-2025-71-3-478-507

Oscillations of a viscous fluid with an inertial free surface
type: item
description: The problem of small motions and normal oscillations of a viscous fluid is investigated. The free surface contains heavy particles of some substance. These particles do not interact with each other during the free surface oscillations, or their interaction is negligible. The original initial-boundary value problem is reduced to the Cauchy problem for a first-order differential equation in a Hilbert space. After a detailed study of the properties of the operator coefficients, a theorem on the solvability of the resulting Cauchy problem is proved. Based on this, sufficient conditions for the existence of a solution to the initial-boundary value problem describing the evolution of the original hydraulic system are found. Statements regarding the structure of the problem spectrum and the basis property of the system of eigenfunctions are proved.
creator: D. O. Tsvetkov
rights: Copyright (c) 2025 Tsvetkov D.O. https://creativecommons.org/licenses/by-nc/4.0
license: https://creativecommons.org/licenses/by-nc/4.0
date: 2025-10-15
publication date: 2025-10-15
volume: 71
number: 3
starting page: 508
ending page: 523
doi: 10.22363/2413-3639-2025-71-3-508-523

Spectral decomposition of self-adjoint operators in Pontryagin and Krein spaces
type: item
description: We consider a self-adjoint operator acting in a Krein space and possessing an invariant subspace that is maximal nonnegative and decomposes into a direct sum of a uniformly positive (i.e., equivalent to a Hilbert space with respect to the inner pseudoscalar product) and a finite-dimensional neutral subspace. We prove the existence of a difference expression that transforms the moment sequence generated by this operator into a sequence representable as the difference of positive moment sequences. In the case of a cyclic operator, this result is applied to construct a function space in which the operator under study is modeled as the operator of multiplication by an independent variable.
creator: V. A. Strauss
rights: Copyright (c) 2025 Strauss V.A. https://creativecommons.org/licenses/by-nc/4.0
license: https://creativecommons.org/licenses/by-nc/4.0
date: 2025-10-15
publication date: 2025-10-15
volume: 71
number: 3
starting page: 524
ending page: 546
doi: 10.22363/2413-3639-2025-71-3-524-546