Pub Date : 2024-07-30DOI: 10.1134/s0012266124040049
V. Yu. Martynova
Abstract
A nonlinear eigenvalue problem for a system of three equations with boundary conditions of the first kind, describing the propagation of electromagnetic waves in a plane nonlinear waveguide, is considered. This is a two-parameter problem with one spectral parameter and a second parameter arising from an additional condition. This condition connects the integration constants that arise when finding the first integrals of the system. The existence of nonlinearizable solutions of the problem is proved.
{"title":"On the Existence of Nonlinearizable Solutions of a Nonclassical Two-Parameter Nonlinear Boundary Value Problem","authors":"V. Yu. Martynova","doi":"10.1134/s0012266124040049","DOIUrl":"https://doi.org/10.1134/s0012266124040049","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A nonlinear eigenvalue problem for a system of three equations with boundary conditions\u0000of the first kind, describing the propagation of electromagnetic waves in a plane nonlinear\u0000waveguide, is considered. This is a two-parameter problem with one spectral parameter and a\u0000second parameter arising from an additional condition. This condition connects the integration\u0000constants that arise when finding the first integrals of the system. The existence of nonlinearizable\u0000solutions of the problem is proved.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"150 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-30DOI: 10.1134/s0012266124040050
V. A. Sadovnichii, Ya. T. Sultanaev, N. F. Valeev
Abstract
We study the statement of the optimization inverse spectral problem with incomplete spectral data for the one-dimensional Schrödinger operator on the entire axis: for a given potential (q_0 ), find the closest function (hat {q} ) such that the first (m ) eigenvalues of the Schrödinger operator with potential (hat {q}) coincide with given values (lambda _k^*in mathbb {R} ), (k={1,dots ,m}).
{"title":"Optimization Inverse Spectral Problem for the One-Dimensional Schrödinger Operator on the Entire Real Line","authors":"V. A. Sadovnichii, Ya. T. Sultanaev, N. F. Valeev","doi":"10.1134/s0012266124040050","DOIUrl":"https://doi.org/10.1134/s0012266124040050","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the statement of the optimization inverse spectral problem with incomplete\u0000spectral data for the one-dimensional Schrödinger operator on the entire axis: for a given\u0000potential <span>(q_0 )</span>, find the closest function <span>(hat {q} )</span> such that the first <span>(m )</span> eigenvalues of the Schrödinger operator\u0000with potential <span>(hat {q})</span> coincide with given values <span>(lambda _k^*in mathbb {R} )</span>, <span>(k={1,dots ,m})</span>.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"361 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-08DOI: 10.1134/s0012266124030078
I. M. Tsvetkov
Abstract
We study a system of equations modeling the dynamic tension of a homogeneous round layer of incompressible perfectly rigid-plastic transversely isotropic material obeying the Mises–Hencky criterion. The upper and lower bases are stress-free, the radial velocity is set on the lateral boundary, and the possibility of thickening or thinning of the layer, simulating formation and further development of a neck, is taken into account. Using the method of asymptotic integration, two characteristic tension modes are identified, that is, relations of dimensionless parameters are determined that necessitate taking into account inertial terms. An approximate solution of the problem is constructed when considering the mode associated with the acceleration on the lateral face reaching its critical values.
{"title":"On the Dynamic Tension of a Thin Round Perfectly Rigid-Plastic Layer Made of Transversely Isotropic Material","authors":"I. M. Tsvetkov","doi":"10.1134/s0012266124030078","DOIUrl":"https://doi.org/10.1134/s0012266124030078","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study a system of equations modeling the dynamic tension of a homogeneous round\u0000layer of incompressible perfectly rigid-plastic transversely isotropic material obeying the\u0000Mises–Hencky criterion. The upper and lower bases are stress-free, the radial velocity is set on the\u0000lateral boundary, and the possibility of thickening or thinning of the layer, simulating formation\u0000and further development of a neck, is taken into account. Using the method of asymptotic\u0000integration, two characteristic tension modes are identified, that is, relations of dimensionless\u0000parameters are determined that necessitate taking into account inertial terms. An approximate\u0000solution of the problem is constructed when considering the mode associated with the acceleration\u0000on the lateral face reaching its critical values.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"49 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-08DOI: 10.1134/s0012266124030017
I. S. Lomov
Abstract
The Sturm–Liouville operator with a singular potential is defined on an interval of the real line. Transmission conditions are specified at an interior point of the interval. The operator potential may have a nonintegrable singularity. For the strong solution of the Cauchy problem for an equation with a parameter, asymptotic formulas and estimates are obtained on each of the solution smoothness intervals.
{"title":"Study of the Asymptotic Properties of the Solution to a Problem with a Parameter for the Sturm–Liouville Operator with a Singular Potential","authors":"I. S. Lomov","doi":"10.1134/s0012266124030017","DOIUrl":"https://doi.org/10.1134/s0012266124030017","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The Sturm–Liouville operator with a singular potential is defined on an interval of the real\u0000line. Transmission conditions are specified at an interior point of the interval. The operator\u0000potential may have a nonintegrable singularity. For the strong solution of the Cauchy problem for\u0000an equation with a parameter, asymptotic formulas and estimates are obtained on each of the\u0000solution smoothness intervals.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"30 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-08DOI: 10.1134/s001226612403008x
A. V. Podobryaev
Abstract
We consider a left-invariant sub-Lorentzian structure on a Lie group. This structure is assumed to be 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 which normal extremal trajectories keep their causal type. Tangent vectors of abnormal extremal trajectories are either lightlike or are tangent vectors of sub-Riemannian abnormal extremal trajectories for the sub-Riemannian distribution spanned by the cone.
{"title":"Sub-Lorentzian Extremals Defined by an Antinorm","authors":"A. V. Podobryaev","doi":"10.1134/s001226612403008x","DOIUrl":"https://doi.org/10.1134/s001226612403008x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider a left-invariant sub-Lorentzian structure on a Lie group. This structure is\u0000assumed to be defined by a closed convex salient cone in the corresponding Lie algebra and a\u0000continuous antinorm associated with this cone. We derive the Hamiltonian system for\u0000sub-Lorentzian extremals and give conditions under which normal extremal trajectories keep their\u0000causal type. Tangent vectors of abnormal extremal trajectories are either lightlike or are tangent\u0000vectors of sub-Riemannian abnormal extremal trajectories for the sub-Riemannian distribution\u0000spanned by the cone.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"10 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-08DOI: 10.1134/s0012266124030066
D. P. Emel’yanov
Abstract
We consider the Dirichlet boundary value problem for an elliptic type equation with irregular noninteger-order degeneration in a rectangle. The coefficients of the differential operator are supposed to be analytic. We construct a formal solution by using the method of spectral separation of singularities in the form of a series; the character of the nonanalytic dependence of the solution on the variable (y) in a neighborhood of (y=0 ) is written out explicitly. We prove the convergence of the series to the classical solution using the Green’s function method.
{"title":"Solution of a Boundary Value Problem for an Elliptic Equation with a Small Noninteger Order Degeneracy","authors":"D. P. Emel’yanov","doi":"10.1134/s0012266124030066","DOIUrl":"https://doi.org/10.1134/s0012266124030066","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the Dirichlet boundary value problem for an elliptic type equation with\u0000irregular noninteger-order degeneration in a rectangle. The coefficients of the differential operator\u0000are supposed to be analytic. We construct a formal solution by using the method of spectral\u0000separation of singularities in the form of a series; the character of the nonanalytic dependence of\u0000the solution on the variable <span>(y)</span> in a neighborhood of\u0000<span>(y=0 )</span> is written out explicitly. We prove the convergence\u0000of the series to the classical solution using the Green’s function method.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"144 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-08DOI: 10.1134/s0012266124030029
A. E. Mamatov
Abstract
In this paper, we study the direct scattering problem on the half-line for the Dirac system of differential equations in the case of finite density with the boundary condition (y_{1}(0)=y_{2}(0) ). The spectrum is studied, the resolvent is constructed, and the spectral expansion in the eigenfunctions of the Dirac operator is obtained.
{"title":"Direct Problem of Scattering Theory for a Dirac System of Differential Equations on the Half-Line in the Case of Finite Density","authors":"A. E. Mamatov","doi":"10.1134/s0012266124030029","DOIUrl":"https://doi.org/10.1134/s0012266124030029","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In this paper, we study the direct scattering problem on the half-line for the Dirac system\u0000of differential equations in the case of finite density with the boundary condition <span>(y_{1}(0)=y_{2}(0) )</span>. The spectrum is studied, the resolvent is\u0000constructed, and the spectral expansion in the eigenfunctions of the Dirac operator is obtained.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"10 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-08DOI: 10.1134/s0012266124030030
E. Mukhamadiev, A. N. Naimov
Abstract
The solvability of a periodic problem for a system of nonlinear second-order ordinary differential equations with a positively homogeneous main part is investigated. New conditions are found that ensure an a priori estimate for the solutions of the periodic problem under consideration. The conditions are stated in terms of the properties of the positively homogeneous main part of the system. Under the a priori estimate, using and developing methods for calculating the mapping degree for vector fields, we prove a theorem on the solvability of the periodic problem that generalizes the results previously obtained by the present authors on the study of the periodic problem for systems of second-order nonlinear ordinary differential equations.
{"title":"On the Solvability of a Periodic Problem for a System of Second-Order Nonlinear Ordinary Differential Equations","authors":"E. Mukhamadiev, A. N. Naimov","doi":"10.1134/s0012266124030030","DOIUrl":"https://doi.org/10.1134/s0012266124030030","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The solvability of a periodic problem for a system of nonlinear second-order ordinary\u0000differential equations with a positively homogeneous main part is investigated. New conditions are\u0000found that ensure an a priori estimate for the solutions of the periodic problem under\u0000consideration. The conditions are stated in terms of the properties of the positively homogeneous\u0000main part of the system. Under the a priori estimate, using and developing methods for calculating\u0000the mapping degree for vector fields, we prove a theorem on the solvability of the periodic problem\u0000that generalizes the results previously obtained by the present authors on the study of the periodic\u0000problem for systems of second-order nonlinear ordinary differential equations.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"150 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-08DOI: 10.1134/s0012266124030108
V. M. Goloviznin, V. V. Konoplyanikov, P. A. Maiorov, S. I. Mukhin
Abstract
The paper is devoted to constructing a numerical algorithm for calculating the blood flow in a volume vessel. A system of differential equations describing the dynamics of fluid in a single vessel with moving walls in cylindrical coordinates is derived assuming axial symmetry in arbitrary Eulerian-Lagrangian variables. A 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.
{"title":"Balance-Characteristic Method for Calculating Hemodynamics of a Single Vessel","authors":"V. M. Goloviznin, V. V. Konoplyanikov, P. A. Maiorov, S. I. Mukhin","doi":"10.1134/s0012266124030108","DOIUrl":"https://doi.org/10.1134/s0012266124030108","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper is devoted to constructing a numerical algorithm for calculating the blood flow\u0000in a volume vessel. A system of differential equations describing the dynamics of fluid in a single\u0000vessel with moving walls in cylindrical coordinates is derived assuming axial symmetry in\u0000arbitrary Eulerian-Lagrangian variables. A balance-characteristic scheme based on the Cabaret\u0000methodology is constructed for the obtained system of equations. The results of calculations of\u0000test problems are given.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"49 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576832","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-08DOI: 10.1134/s0012266124030042
M. V. Shamolin
Abstract
Tensor invariants (first integrals and differential forms) of homogeneous dynamical systems on the tangent bundles of 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 needed for the integration of geodesic, potential, and dissipative systems is shown. At the same time, the force fields introduced make the systems in question dissipative with dissipation of different signs and generalize the previously considered ones.
{"title":"Invariants of Geodesic, Potential, and Dissipative Systems with Three Degrees of Freedom","authors":"M. V. Shamolin","doi":"10.1134/s0012266124030042","DOIUrl":"https://doi.org/10.1134/s0012266124030042","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Tensor invariants (first integrals and differential forms) of homogeneous dynamical systems\u0000on the tangent bundles of smooth three-dimensional manifolds (systems with three degrees of\u0000freedom) are presented in this paper. The connection between the presence of such invariants and\u0000the complete set of the first integrals needed for the integration of geodesic, potential, and\u0000dissipative systems is shown. At the same time, the force fields introduced make the systems in\u0000question dissipative with dissipation of different signs and generalize the previously considered\u0000ones.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"31 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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