Том 27, № 2 (2025)
- Жылы: 2025
- ##issue.datePublished##: 30.06.2025
- Мақалалар: 7
- URL: https://journals.rcsi.science/2079-6900/issue/view/21790
- DOI: https://doi.org/10.15507/2079-6900.27.202502
Бүкіл шығарылым
Mathematics
On orthogonalization of Schoenberg splines
Аннотация
The article is devoted to the application of the author's orthogonalization procedure of finite functions, which does not destroy their finite supports, to Schoenberg splines of the third degree. A general algorithm for modifying the Schoenberg mother spline within the framework of this orthogonalization procedure is described. It is shown that orthogonalization of the grid set of splines generated by the Schoenberg spline is achieved without changing the finite supports of the splines in the case of using eight step functions to modify the mother spline. Sixteen variants of orthogonalization for the cubic Schoenberg splines by step functions are found. In the first group of eight variants, all coefficients of the modifying step functions have real values, but the Schoenberg splines after such modification are not even or odd functions. In each of the eight variants of the second group, two coefficients are complex, and the remaining six coefficients have real values. The modified Schoenberg splines of the second group are sums of even and odd functions. A theorem on the order of approximation of any function from the Sobolev space by linear combinations of constructed orthogonal Schoenberg splines is proved.
Middle Volga Mathematical Society journal. 2025;27(2):111-126
111-126
About an algorithm for solving the speed problem in linear systems with convex restrictions on phase variables and control
Аннотация
The problem optimal speed control is investigated in the case when the process is described by a system of linear ordinary differential equations with nonlinear convex restrictions on phase variables and control. By moving from n-dimensional Euclidean space to Hilbert space, the optimal control problem with restrictions on phase variables and control is reduced to an optimal speed problem without restrictions. It is shown that the reachability region in the new space is a convex set. To solve the resulting problem, a modified method of separating hyperplanes is used. One of the key points of this method, on which the convergence speed of the algorithm depends, is finding the normal to the separating hyperplane. In this work, this normal at each iteration is constructed by minimizing a distance-type functional on the convex hull of points supporting the reachability set obtained at previous iterations. After finding the normal to the separating hyperplane, a hyperplane supporting the reachable region is constructed, which is then continuously transferred in increasing time and the first moment in time is found at which the supporting hyperplane reaches the given end point. This moment is taken as the next approximation to the performance time. A theorem is formulated on the convergence of successive approximations in time to the value of the performance time and on the weak convergence of a sequence of controls to an optimal control. The algorithm is tested by solving the problem of external heating of an unlimited plate to a given temperature in a minimal time, taking into account restrictions on tensile and compressive thermal stresses. The results of a computational experiment are presented.
Middle Volga Mathematical Society journal. 2025;27(2):127-142
127-142
A Study of Numerical Methods for Solving the Nonlinear Energy Resources Supply-Demand System
Аннотация
n this study, we implement and estimate various numerical methods for solving a nonlinear differential equation system modeling energy resources supply-demand dynamics. Both single-step methods (Taylor series, Runge-Kutta) and multi-step methods (Adams – Bashforth, Adams Predictor-Corrector) are employed. In addition to standard fourth-order methods, higher-order techniques such as the fifth-order Runge-Kutta method and the sixth- order Taylor series method are also applied. Furthermore, along with fixed-step numerical methods, we implement and assess adaptive step-size methods, including the explicit Runge- Kutta method of order 5(4) (that is RK45), the explicit Runge-Kutta method of order 8(5,3) (or DOP853), the implicit Runge-Kutta method from the Radau IIA family of order 5 (Radau), the implicit method based on backward differentiation formulas (BDF), and the Adams/BDF method with automatic switching (LSODA). The results indicate that, in the cases we considered, single-step methods are more effective than multi-step ones in capturing and tracking rapid variations of the system, while multi-step methods require less computation time. Adaptive step-size numerical methods demonstrate both flexibility and stability. Through the evaluation and analysis of numerical solutions obtained by various methods, the behaviour and dynamic characteristics of the system are explored.
Middle Volga Mathematical Society journal. 2025;27(2):143-170
143-170
Rotation sets of SO(3)-extensions of quasiperiodic flows
Аннотация
In this paper, we construct a class of special flows on a multidimensional torus and a topological invariant of such flows, i.e. a rotation set. Such flows arise while reducing linear systems of differential equations with quasiperiodic coefficients to a triangular form. In the process of such a reduction, we obtain a system of nonlinear differential equations on a multidimensional torus, which generates a projective flow induced by the original linear system. In this paper, we use known results from the theory of matrix groups and Lie algebras and construct an algorithm for SO$(n)$-extension of a quasiperiodic linear system. The resulting system of equations admits a reduction in order, which allows us to write the right-hand sides as trigonometric polynomials in Euler angles on a sphere. The case $n=3$ is considered separately. The equations defining the projective flow are written explicitly. The projective flow is defined on a torus of dimension $m+2$, where $m$ is the dimension of the original torus. The structure of this flow is determined by topological invariants of the flow. For example, a non-singular flow on a two-dimensional torus has a topological invariant - the rotation number (A. Poincare). Using M. Herman's method, it is possible to prove the existence and uniqueness of the rotation vector $(\rho_1,\rho_2)$ for the projective flow on $\T^{m+2}$. Using S. Schwartzman's theory defining the rotation set for flows on compact metric spaces, it is shown that the component $\rho_2=0$. Here, the fact is used that the dimension of the maximal toric subalgebra of the algebra so$(3)$ is equal to one.
Middle Volga Mathematical Society journal. 2025;27(2):171-184
171-184
Planarity ranks of semigroup varieties generated by semigroups of order four
Аннотация
This paper classifies semigroup varieties generated by fourth-order semigroups according to their planarity ranks. The aim of the study is to establish a complete list of possible values of planarity ranks and to identify the main factors determining the possibility of planar stacking of Cayley graphs of free semigroups of the considered varieties. Methods from graph theory and algebras of identities are applied, using innovative algorithmic approaches to verify equality via the automated proof systems Prover9 and Mace4. The existing flat graph stackings for the Cayley graphs of the semigroups under consideration are shown in the figures. If there is no planarity, the particular forbidden minor discovered is indicated: a complete fifth-order graph or a complete bipartite graph containing three vertices in each of the parts. Special attention is paid to the statistical processing of the obtained results by the principal components analyse and the construction of hierarchical clustering. The figures show hierarchical trees, factor planes, correlation circles, and column diagrams of general inertia decomposition along coordinate axes. Although the planarity of the Cayley graph for a free semigroup of a manifold was previously intuitively associated with the complexity degree of the defining identities, in this paper this dependence is for the first time given a rigorous quantitative expression, depicted in tables. Within the framework of the study, auxiliary parameters are introduced, which allows to significantly increase the explanatory power of the model and divide manifolds into groups according to topological characteristics. As a result of the analysis it is established that the leading factors influencing the value of ranks are the parameters reflecting the differences of positions of the symbol «z» in the basis identities.
Middle Volga Mathematical Society journal. 2025;27(2):185-228
185-228
Asymptotic and numerical study to the damped Schamel equation
Аннотация
Analytical and numerical solutions of the damped Schamel equation, describing the dynamics of ion-acoustic waves in magnetized plasma, are presented. A small parameter is introduced in the equation before the dissipative term, ensuring that in its absence the solution reduces to a solitary wave (soliton). The asymptotic method employed for solving the equation is a variant of the Krylov-Bogolyubov-Mitropolsky multiple-scale technique. In the first-order approximation, the solution is described by a traveling solitary wave with slowly varying parameters. The second-order approximation yields the evolution laws for the soliton’s amplitude and phase as functions of «slow» time. Additionally, exact integral conservation laws (mass and energy of the wave field), derived directly from the original damped Schamel equation, are utilized. These integrals allow estimating the soliton’s radiative losses, particularly the mass of the so-called tail formed behind the soliton due to dissipation. Direct numerical solutions of the original equation, obtained via a pseudospectral method, confirm the asymptotic laws governing the soliton’s amplitude decay caused by dissipation. Another limiting case - strong dissipation (dominant over nonlinearity and dispersion), is also investigated, demonstrating that the soliton decays as a linear impulse, which is validated numerically.
Middle Volga Mathematical Society journal. 2025;27(2):229-242
229-242
Mathematical modeling and computer science
A Numerical Algorithm for Studying Subsonic Chemically Reacting Flow in the Presence of Laser Radiation
Аннотация
The article presents a numerical algorithm for studying subsonic viscous chemically active flows in the presence of laser radiation. The process model is described in the Navier – Stokes approximation adjusted for the subsonic flow regime, with addition of source terms corresponding to chemical transformations. An additional ordinary differential equation describing the propagation of laser radiation along the length of the region under study is introduced as well. The computational algorithm is based on splitting by physical processes. This makes it possible to calculate separately changes in concentrations during chemical transformations, convective fluxes, dissipative terms, dynamic pressure deviation and propagation of laser radiation. To account for the dissipative terms (diffusion, viscosity, and thermal conductivity), the local iteration method based on Chebyshev polynomials’ ordering. Due to the possible use of a larger total calculation time step, the software implementation of the constructed algorithm reveals shorter calculation times using the local iteration method for calculating dissipative terms in comparison with the algorithm calculating them based on a scheme with central differences.
The algorithm was verified using the example of methane conversion by comparing it with the calculation of the stoichiometric balance of the brutto-reaction, as well as by studying the convergence of the solution on a sequence of thickening grids. Based on the developed algorithm, a numerical study of non-oxidative conversion of methane under the influence of laser radiation in a circular tube was carried out, and graphs of the distribution of the main characteristics of the mixture were obtained.
Middle Volga Mathematical Society journal. 2025;27(2):243-254
243-254