Vol 237 (2024)
Статьи
On the Solvability of a Variational Parabolic Equation with a Non-Local-in-Time Condition on the Solution
Abstract
In a separable Hilbert space, a parabolic equation with a special time-weighted integral condition is considered. Conditions are obtained under which the solution to the problem is more smooth than a weak solution, the existence and uniqueness of which was proved earlier.
3-9
The Cauchy Problem with a Parameter Perturbed by a Linear Functional
Abstract
In this paper, we consider a Cauchy problem with a parameter perturbed by a linear functional. For any value of the parameter, the problem has a trivial solution. We obtain necessary and sufficient conditions for values of the parameter such that in their neighborhoods non-trivial solutions in the class of real continuous functions exist. A method of constructing such solutions is proposed.
10-17
Local Bifurcations in One Version of the Multiplier-Accelerator Model
Abstract
The well-known mathematical model of macroeconomics “multiplier-accelerator” is considered in a nonlinear version with spatial factors. We study two versions of the corresponding boundary-value problem. In the first version, where the spatial dissipation is significant in the linear statement, the boundary-value problem has limit cycles that arise as a result of Andronov–Hopf bifurcations. The second version of the boundary-value problem arises when dissipation in the linear formulation is neglected. In this weakly dissipative version, the boundary-value problem has a countable set of finite-dimensional cycles and tori. All such invariant manifolds are unstable. The analysis of the problem is based on methods of the theory of infinite-dimensional dynamic systems.
18-33
Converting a Continuous Fuzzy Signal by a Linear Dynamic System
Abstract
In this paper, we apply the method of Green’s function to the search for bounded solutions of a high-order linear differential equation with constant coefficients and a fuzzy right-hand side. A class of equations with positive coefficients and a nonnegative Green’s function is distinguished, for which the results on the existence and smoothness of a fuzzy solution bounded on the whole axis are established. We prove that in the case where the right-hand side has a triangular form, the solution has the same form. Examples of radio engineering circuits with fuzzy input signals are considered.
34-48
Invariants of Homogeneous Dynamic Systems of Arbitrary Odd Order with Dissipation. II. Fifth-Order Systems
Abstract
In this paper, we present new examples of integrable dynamical systems of the fifth 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.
49-75
Training a Neural Network for a Hyperbolic Equation by Using a Quasiclassical Functional
Abstract
We study the problem of constructing a loss functional based on the quasiclassical variational principle for training a neural network, which approximates solutions of a hyperbolic equation. Using the method of symmetrizing operator proposed by V. M. Shalov, for the secondorder
hyperbolic equation, we construct a variational functional of the boundary-value problem, which involves integrals over the domain of the boundary-value problem and a segment of the boundary, depending on first-order derivatives of the unknown function. We demonstrate that the neural network approximating the solution of the boundary-value problem considered can be trained by using the constructed variational functional.
76-86