Contemporary Mathematics. Fundamental Directions
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)
Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)
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Second-order difference scheme for hyperbolic equations with unbounded delay
The present paper is devoted to the study the initial value problem for the hyperbolic equation with unbounded time delay term \( \begin{equation*} \begin{cases} \dfrac{d^{2}v(t)}{dt^{2}}+A^{2}v(t)=a\left( \dfrac{dv(t-\omega )}{dt} +Av(t-\omega )\right) +f(t), & t>0, \\ v(t)=\varphi (t), & -\omega \leq t\leq 0 \end{cases} \end{equation*} \) in a Hilbert space H with a self-adjoint positive definite operator A. The second order of accuracy difference scheme for the numerical solution of the differential problem is presented. The main theorem on stability estimates for the solutions of this difference scheme is established. In practice, the stability estimates for solutions of four problems for hyperbolic difference equations with time delay are proved.
Copyright (c) 2025 Ashyralyev A. https://creativecommons.org/licenses/by-nc/4.0
A mathematical theoryof the accelerated expansion of the Universe based on the principle of least action
In classical works, the equations for gravitational and electromagnetism fields are proposed without deriving the right-hand sides. Here, we derive the right-hand sides and analyze the energy-momentum tensor within the framework of the Vlasov-Maxwell-Einstein equations and consider cosmological models such as Milne-McCrea and Friedmann. This allows us to place General Relativity (GR) on a rigorous mathematical foundation: to derive a closed system of GR equations from the principle of least action and provide a rigorous definition of cosmological solutions. This explains the accelerated expansion of the Universe without Einstein’s lambda, dark energy, or fantastic new fields, but as a simple relativistic effect.
Copyright (c) 2025 Vedenyapin V.V., Batishcheva Y.G., Goryunova M.V., Russkov A.A. https://creativecommons.org/licenses/by-nc/4.0
Parameterization of control functions in the problem of modeling HIV infection therapy
Mathematical modeling is actively used to study the mechanisms of human immunodeficiency virus of type 1 (HIV-1) infection. Current HIV-1 therapy involves the regular, lifelong use of multiple antiviral drugs. However, this therapy is associated with varying degrees of side effects due to toxicity, drug interactions, resistance development, and high cost. Mathematical models of HIV-1 infection and optimal control methods can be used to develop effective regimens for applying multiple antiretroviral drugs, taking into account the immune status of HIV-1-infected patients. In this study, we identify the pharmacodynamic parameters of drugs based on a previously constructed stochastic model of the processes that determine viral replication in infected cells. We also study the efficiency of standard therapy for various HIV-1 infection regimens using a system dynamics model. The results of the study indicate the need to take into account differences in the body’s response to therapy based on the criterion of efficiency, which actualizes the task of selecting individual therapy regimens using optimal control methods based on physiologically approved models of HIV-1 infection.
Copyright (c) 2025 Grebennikov D.S., Lyfenko A.I., Timokhin A.M., Savinkov R.S., Bocharov G.A. https://creativecommons.org/licenses/by-nc/4.0
A study of the solvability of the Bingham alpha model
This article studies the solvability of the initial-boundary value problem for the alpha model of a Bingham-type viscoplastic fluid with periodic conditions on the spatial variables. Using an approximation-topological approach, we prove the existence of weak solutions to the alpha model under study and establish the convergence of the alpha model solutions to the solutions of the original model as the alpha parameter tends to zero.
Copyright (c) 2025 Zvyagin A.V., Tolstoy N.V. https://creativecommons.org/licenses/by-nc/4.0
On the differential model of sandpiles growing in a silo
We discuss some features of a boundary value problem for a system of PDEs that describes the growth of a sandpile in a container under the action of a vertical source. In particular, we characterize the long-term behavior of the profiles, and we provide a sufficient condition on the vertical source that guarantees the convergence to the equilibrium in a finite time. We show by counterexamples that a stable configuration may not be reached in a finite time, in general, even if the source is timeindependent. Finally, we provide a complete characterization of the equilibrium profiles.
Copyright (c) 2025 Crasta G., Malusa A. https://creativecommons.org/licenses/by-nc/4.0
On the damping of a neutral-type control system on a temporal star graph with time-proportional delay
On a temporal star graph, we consider the problem of optimally damping a control system for a generalized pantograph equation, which is a neutral-type equation with time-proportional delay. The delay in the system propagates through an internal vertex of the graph. We study the variational problem of minimizing the energy functional, taking into account the probabilities of scenarios corresponding to different edges. We establish that the optimal trajectory satisfies Kirchhofftype conditions at the internal vertex. We prove the equivalence of the variational problem to a certain boundary-value problem for second-order functional differential equations on the graph, and establish the unique solvability of both problems.
Copyright (c) 2025 Lednov A.P. https://creativecommons.org/licenses/by-nc/4.0
For Hamiltonian systems on symplectic manifolds with constraints in the Dirac model of generalized Hamiltonian dynamics, V. V. Kozlov considered the operation of symplectic projection of a Hamiltonian vector field for the case of generalized nonintegrable differential constraints. This paper considers a constraint regularization method that circumvents the degeneracy of the symplectic projection operation in the case of an odd number of constraints. The method is based on embedding of the original system into an extended system of higher dimension with an increased number of constraints.
Copyright (c) 2025 Salnikova T.V., Kugushev E.I. https://creativecommons.org/licenses/by-nc/4.0
A class of anisotropic diffusion-transport equations in nondivergent form
We generalize Einstein’s probabilistic method for the Brownian motion to study compressible fluids in porous media. The multi-dimensional case is considered with general probability distribution functions. By relating the expected displacement per unit time with the velocity of the fluid, we derive an anisotropic diffusion equation in non-divergence form that contains a transport term. Under the Darcy law assumption, a corresponding nonlinear partial differential equations for the density function is obtained. The classical solutions of this equation are studied, and the maximum and strong maximum principles are established. We also obtain exponential decay estimates for the solutions for all time, and particularly, their exponential convergence as time tends to infinity. Our analysis uses some transformations of the Bernstein-Cole-Hopf type which are explicitly constructed even for very general equation of state. Moreover, the Lemma of Growth in time is proved and utilized in order to achieve the above decaying estimates.
Copyright (c) 2025 Hoang L., Ibragimov A.I. https://creativecommons.org/licenses/by-nc/4.0
Dynamics of the conformation tensor in viscoelastic FENE polymer models
In this work, the equations for the dynamics of the invariants of the conformational tensor for FENE polymer solution models are derived and integrated. Explicit formulas for the invariants as functions of the time parameter along the trajectory of fluid particles are obtained. The invariants are represented as functions of the Lambert function. A description of the qualitative behavior of the invariants under different regimes is given.
Copyright (c) 2025 Chupakhin A.P., Stetsyak E.S., Chutkov D.S. https://creativecommons.org/licenses/by-nc/4.0