Vol 243 (2025)
Articles
Revaz Valerianovich Gamkrelidze
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;243:3-4
3-4
Application of Picard's method to Cauchy problem solution to some fractional differential equations
Abstract
In this paper, we apply the Picard method for solving the Cauchy problem for some fractional differential equations with Atangana–Baleanu fractional derivative. An iterative scheme is derived and its convergence is proved.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;243:5-10
5-10
Contrast structures in a reaction-diffusion system with multiscale diffusion coefficients and discontinuous reaction functions
Abstract
In this paper, we examine a one-dimensional reaction-diffusion system with different-scale diffusion coefficients, discontinuous reaction functions, and Neumann boundary conditions. We demonstrate that a singular perturbation in the fast-component equation and reaction discontinuities lead to the formation of contrast structures with internal transition layers. Also, we analyze the existence, uniqueness, and asymptotic stability of stationary solutions. The obtained results provide theoretical justification for numerical methods applicable to such systems and enable prediction of behavior of solutions in domains of sharp gradients, which is crucial for developing efficient computational algorithms.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;243:11-24
11-24
Periodic traveling waves of the Kuramoto–Sivashinsky equation
Abstract
A periodic boundary-value problem for the Kuramoto–Sivashinsky equation is considered. We prove that there exists a two-parameter family of traveling-wave solutions and obtain asymptotic formulas for them. We also prove that the set of such solutions forms a two-dimensional invariant manifold, which is a local attractor. The indicated solutions have different periods in the variable $t$, are unstable in the Lyapunov sense, but are stable in the Perron, Poincare, and Zhukovsky senses. The study is based on the theory of invariant manifolds and Poincare–Dulac normal forms.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;243:25-37
25-37
Formation of a boundary-layer solution in a problem for a system of reaction-diffusion equations in a limited volume
Abstract
We consider a system of equations that describes a two-component chemical reaction in a limited volume. The reaction is assumed to occur in a solution, the concentration of reaction products increases in time, then becomes maximal possible under the given conditions (i.e., saturation occurs), and then the reaction terminates. A similar formulation can be used for describing microscopic processes occurring when CO$_2$ is injected into a rock, which is a porous medium with pores filled with water. For a system of two equations of the “reaction-diffusion” type on a segment, we show that in a finite time, a solution close to a stationary distribution corresponding to the concentration of a saturated solution under given conditions is formed from a given initial function.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;243:38-44
38-44
On periodic solutions with boundary layers in problems with nonlinear singular boundary conditions
Abstract
For a reaction-diffusion problem with nonlinear singularly perturbed boundary conditions, we prove the existence and examine the stability of periodic solutions possessing boundary layers. Conditions of the asymptotic stability of these solutions in the Lyapunov sense are obtained. The proof is based on the asymptotic method of differential inequalities.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;243:45-55
45-55
Front formation in the reaction-diffusion problem with nonlinear diffusion
Abstract
The process of the formation of a front-type solution in the reaction-diffusion equation in the case of spatially inhomogeneous nonlinear diffusion is considered. Sufficient conditions are formulated to ensure the formation of a sharp inner transition layer (front) in a neighborhood of a certain point, and estimates of the duration of the transition process are obtained.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;243:56-62
56-62
On nonnegative solutions of systems of linear differential equations with variable coefficients under fuzzy initial data and inhomogeneities
Abstract
Systems of linear differential equations with variable coefficients with fuzzy initial data and inhomogeneities are examined. In the case of matrices with nonnegative elements, we prove the existence of strong solutions of homogeneous and inhomogeneous initial problems and periodic solutions. The main attention is paid to the nonnegativity of solutions to the corresponding linear problems. It is established under the additional assumption of nonnegativity of fuzzy initial data and heterogeneities. An application to a dynamic model of input-output balance with fuzzy data is considered.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;243:63-77
63-77
Normality in real AW*-algebras
Abstract
This paper is devoted to the study of the concept of normality in real AW*-algebras. The authors examine whether normality is preserved when passing from a complex AW*-algebra to its real part and prove that all real AW*-factors are normal. Conditions are established under which a real AW*-algebra is normal, in particular, when its center is locally $\sigma$-finite. The obtained results serve as real analogs of known theorems on normality in AW*-algebras and contribute to the development of operator algebra theory in the real setting.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;243:78-80
78-80
Features of the moving-front solution for a two-dimensional problem with a discontinuous cubic nonlinearity
Abstract
In this paper, we examine solutions of the moving-front-type for a two-dimensional reaction-diffusion equation with cubic nonlinearity and propose a method for obtaining asymptotic approximations of moving fronts propagating in a medium with discontinuous characteristics is presented. The basic features arising in solving the two-dimensional problem are discussed.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;243:81-89
81-89
Control of dynamic systems under information deficit. Guaranteed approach. I. Estimation algorithms
Abstract
In this paper, we consider the problems of synthesis of positional control for linear dynamic systems in the case where it is necessary to guarantee the achievement of the control goal, and the disturbances acting on the dynamic system and the interference in the information channels of the system are known with an accuracy of sets in which they can take any values. We construct information sets and forecast sets that contain the state vector. Control problems are solved for the case of specifying requirements for the system in the form of sets in the phase space to which the state vector must belong, taking into account the constraints on control, or with requirements in the form of a quadratic functional. The application of Lyapunov functions for control synthesis is shown. The first part of this work is devoted to evaluation algorithms.
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory. 2025;243:90-112
90-112