Vol 71, No 4 (2025)

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.

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.

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.

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.

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.