Vol 240 (2025)
Articles
Optimal boundary control of oscillations of a string with given intermediate values of the speed for minimizing boundary energy
Abstract
For the equation of oscillation of a string with given initial and terminal conditions, we consider optimal boundary control problems with given intermediate conditions on the values of the speed with criteria for the qualities of the integrals of the boundary energies. The control is perform by displacements of ends of the string. Boundary energy integrals are considered over the entire time interval. We propose a constructive approach to constructing optimal boundary controls for oscillations based on the methods of separation of variables and moment problems. A computational experiment was carried out and the results obtained were analysed.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;240:3-18
3-18
Exact and approximate solutions to the quasilinear parabolic system “predator-prey” with zero fronts
Abstract
In this paper, we consider the second-order quasilinear parabolic system known in population biology as the predator-prey model and examine exact and approximate solutions with two zero fronts on which at least one of two unknown functions vanish; both these functions are positive between the fronts. We search for exact solutions in the form of polynomials in powers of the spatial variable with the coefficients depending on time. To construct approximate solutions, we propose a numerical algorithm, which is a combination of the collocation method based on the expansion of the right-hand sides by the radial basis functions and the finite-difference approximation of the derivatives in time. The algorithm is verified by model examples; the results obtained are consistent with the exact solutions found.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;240:19-28
19-28
Classical scattering matrix for hard and soft excitations in a plasma with non-abelian interaction
Abstract
Within the framework of the Zakharov–Shulman approach, we determine the classical scattering matrix for the simplest process of interaction between hard and soft excitations in a quark-gluon plasma. Calculations are performed in close analogy with the methods of quantum field theory, with the replacement of the quantum commutator of quantum field operators by the so-called Lie–Poisson bracket of classical variables. The classical $\mathcal{S}$-matrix is determined in the form of the most general integro-power series in asymptotic values of the normal bosonic variables $c^{a}_{\boldsymbol{k}}(t)$ and $c^{\ast a}_{\boldsymbol{k}}(t)$ describing the soft gluon excitations of the system and the color charge $\mathcal{Q}^{a}(t)$ of the hard particle at $t\rightarrow\infty$. The first nontrivial contribution to the given $\mathcal{S}$-matrix is obtained.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;240:29-38
29-38
On the solvability and limiting properties of some systems of partial differential equations with a small parameter in the principal part
Abstract
In this paper, we consider linear systems of partial differential equations involving a small parameter as the coefficient of one of higher derivatives and establish a relationship between solutions of the singularly perturbed problem and solutions of the limit system in which the perturbation parameter is equal to zero. We examine the influence of the matrix pencil composed of the coefficients of the equations on the solvability of both original and limit problems and state sufficient conditions for the passage to the limit in terms of the parameter from the perturbed system to the limit system. Using vector-matrix methods, we obtain explicit formulas for solutions of the problems considered.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;240:39-48
39-48
Invariants of homogeneous dynamic systems of arbitrary odd order with dissipation. V. General case
Abstract
In this paper, we present new examples of integrable dynamical systems of any odd order that are homogeneous in part of the variables. In these systems, subsystems on the tangent bundles of lower-dimensional manifolds can be distinguished. In the cases considered, the force field is partitioned into an internal (conservative) part and an external part. The external force introduced by a certain unimodular transformation has alternate dissipation; it is a generalization of fields examined earlier. Complete sets of first integrals and invariant differential forms are presented.The first part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 236 (2024), pp. 72–88.The second part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 237 (2024), pp. 49–75.The third part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 238 (2024), pp. 69–100.The fourth part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 239 (2024), pp. 62–97.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;240:49-89
49-89