Vol 60, No 3 (2024)
Articles
STUDY OF THE ASYMPTOTIC PROPERTIES OF THE SOLUTION TO A PROBLEM WITH A PARAMETER FOR THE STURM–LIOUVILLE OPERATOR WITH A SINGULAR POTENTIAL
Abstract
The Sturm–Liouville operator with a singular potential on an interval with conjugation conditions at the interior point of the interval is considered. The operator potential may have a non-integrable singularity. For a strong solution of the Cauchy problem for an equation with a parameter, asymptotic formulas and estimates are obtained on each of the smoothness segments of the solution.
Differential Equations. 2024;60(3):291-297
291-297
DIRECT PROBLEM OF SCATTERING THEORY FOR A SYSTEM OF DIRAC DIFFERENTIAL EQUATIONS ON THE SEMI-AXIS IN THE CASE OF FINITE DENSITY
Abstract
In this paper, we study the direct scattering problem on the semi-axis for the system of Dirac differential equations in the case of finite density with the boundary condition y1(0) =y2(0). The spectrum was studied, the resolvent and spectral expansion in terms of eigenfunctions of the Dirac operator were constructed.
Differential Equations. 2024;60(3):298-311
298-311
ON THE SOLVABILITY OF A PERIODIC PROBLEM FOR A SYSTEM OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER
Abstract
In this paper is investigated the solvability of a periodic problem for a system of nonlinear ordinary differential equations second order with the main positively homogeneous part. New conditions have been found that provide an a priori estimate solutions of the periodic problem under consideration. The conditions for the a priori estimate are formulated in terms of the properties of the main positively homogeneous part of the system of equations. Under the conditions of an a priori estimate, using and developing methods for calculating the mapping degree, a theorem on the solvability of the periodic problem is proven. The proven theorem generalizes previously obtained the authors’ results on the study of a periodic problem for systems of nonlinear ordinary differential equations of the second order.
Differential Equations. 2024;60(3):312-321
312-321
INVARIANTS OF GEODESIC, POTENTIAL AND DISSIPATIVE SYSTEMS WITH THREE DEGREES OF FREEDOM
Abstract
Tensor invariants (first integrals and differential forms) of homogeneous dynamical systems on tangent bundles to smooth three-dimensional manifolds (systems with three degrees of freedom) are presented in this paper. The connection between the presence of such invariants and the complete set of the first integrals, which are necessary for the integration of geodesic, potential, and dissipative systems, is shown. At the same time, the introduced force fields make the considered systems dissipative with dissipation of different signs and generalize the previously considered ones.
Differential Equations. 2024;60(3):322-345
322-345
ON SOLVABILITY OF INITIAL AND BOUNDARY VALUE PROBLEMS FOR ABSTRACT FUNCTIONAL-DIFFERENTIAL EULER–POISSON–DARBOUX EQUATIONS
Abstract
In the Banach space for functional-differential equation, generalizing the Euler–Poisson–Darboux equation, the Cauchy problem and boundary value problems of Dirichlet and Neumann are consider. A sufficient condition for solvability is proved Cauchy problem and an explicit form of the resolving operator is indicated, which is written using the ones introduced by the author Bessel and Struve operator functions. For boundary value problems in the hyperbolic case, sufficient conditions for their unique solvability imposed on the operator coefficient of the equation and boundary elements.
Differential Equations. 2024;60(3):346-364
346-364
SOLUTION OF A BOUNDARY PROBLEM FOR AN ELLIPTIC EQUATION WITH A SMALL NONINTEGER ORDER DEGENERACY
Abstract
We consider a Dirichlet boundary problem for an elliptic type equation with a non-regular degeneracy of noninteger order in a rectangle. The coefficients of the differential operator are supposed to be analytic. We build a formal solution by using the method for spectral separation of the singularities. The solution is series where its non-analytic dependency on ???? near point ???? = 0 is written explicitly. We proof the convergence of the series to the classical solution using the Green’s function method.
Differential Equations. 2024;60(3):365-374
365-374
ON THE DYNAMIC STRETCHING OF A THIN ROUND IDEALLY RIGID PLASTIC LAYER MADE OF A TRANSVERSELY ISOTROPIC MATERIAL
Abstract
A system of equations modeling the dynamic stretching of a homogeneous circular layer of incompressible ideally rigid-plastic transversely isotropic material obeying the Mises–Hencky criterion is studied. The upper and lower bases are stress-free, the radial velocity is set at the lateral boundary, and the possibility of thickening or thinning of the layer is taken into account, which simulates neck formation and further development of the neck. Using the method of asymptotic integration, two characteristic stretching modes are identified, that is, the relations of dimensionless parameters are determined, in which consideration of inertial terms is necessary. When considering the regime associated with the achievement of acceleration on the side face of its critical values, an approximate solution of the problem was constructed.
Differential Equations. 2024;60(3):375-385
375-385
SUB-LORETZIAN EXTREMALS DEFINED BY AN ANTINORM
Abstract
We consider a left-invariant sub-Lorentzian structure on a Lie group. We assume that this structure is defined by a closed convex salient cone in the corresponding Lie algebra and a continuous antinorm associated with this cone. We derive the Hamiltonian system for sub-Lorentzian extremals and give conditions under that normal extremal trajectories keep their causal type. Tangent vectors of abnormal extremal trajectories are either light-like or tangent vectors of sub-Riemannian extremal trajectories for the sub-Riemannian distribution spanned by the cone.
Differential Equations. 2024;60(3):386-398
386-398
ON EXACT GLOBAL CONTROLLABILITY OF A SEMILINEAR EVOLUTIONARY EQUATION
Abstract
For a Cauchy problem associated with a controlled semilinear evolutionary equation with optionally bounded operator in a Hilbert space we obtain sufficient conditions for exact controllability to a given final state (and also to given intermediate states at intermediate time moments) on a arbitrarily fixed (without additional conditions) time interval. Here we use the Minty–Browder’s theorem and also a chain technology of successive continuation of the solution to a controlled system to intermediate states. As examples we consider a semilinear pseudoparabolic equation and a semilinear wave equation.
Differential Equations. 2024;60(3):399-417
399-417
BALANCE-CHARACTERISTIC METHOD FOR CALCULATING HEMODYNAMICS OF A SINGLE VESSEL
Abstract
The paper is devoted to the construction of a numerical algorithm for calculating the blood flow in a volume vessel. The derivation of the system of differential equations describing the dynamics of fluid in a single vessel with moving walls in cylindrical coordinates assuming axial symmetry in arbitrary eulerianlagrangian variables is given. Balance-characteristic scheme based on the CABARET methodology is constructed for the obtained system of equations. The results of calculations of test problems are given.
Differential Equations. 2024;60(3):418-432
418-432