Pub Date : 2024-09-19DOI: 10.1134/s0012266124060077
V. I. Golubev, I. S. Nikitin, A. V. Shevchenko, I. B. Petrov
Abstract
The paper examines the dynamic behavior of elastoviscoplastic media under the action of an external load. For the case of a linear viscosity function and a nonlinear softening function, an explicit–implicit calculation scheme is constructed that permits one to obtain a numerical solution of the original semilinear hyperbolic problem. This approach does not involve the use of the method of splitting into physical processes. Despite this, an explicit computational algorithm is obtained that can be effectively implemented on modern computing systems.
{"title":"Explicit–Implicit Schemes for Calculating Dynamics of Elastoviscoplastic Media with Softening","authors":"V. I. Golubev, I. S. Nikitin, A. V. Shevchenko, I. B. Petrov","doi":"10.1134/s0012266124060077","DOIUrl":"https://doi.org/10.1134/s0012266124060077","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper examines the dynamic behavior of elastoviscoplastic media under the action of\u0000an external load. For the case of a linear viscosity function and a nonlinear softening function, an\u0000explicit–implicit calculation scheme is constructed that permits one to obtain a numerical solution\u0000of the original semilinear hyperbolic problem. This approach does not involve the use of the\u0000method of splitting into physical processes. Despite this, an explicit computational algorithm is\u0000obtained that can be effectively implemented on modern computing systems.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252303","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 : 2024-09-19DOI: 10.1134/s0012266124060028
Y. Li, A. A. Grin, A. V. Kuzmich
Abstract
New methods for constructing two Dulac–Cherkas functions are developed using which a better, depending on the parameter (lambda >0), inner boundary of the Poincaré–Bendixson annulus (A(lambda ) ) is found for the Rayleigh system. A procedure is proposed for directly finding a polynomial whose zero level set contains a transversal oval used as the outer boundary of (A(lambda )). An interval for (lambda ) is specified with which the best outer boundary of the annulus ( A(lambda )) is a closed contour composed of two arcs of the constructed oval and two arcs of unclosed curves of the zero level set of one of the Dulac–Cherkas functions. Thus, a refined global Poincaré–Bendixson annulus for the limit cycle of the Rayleigh system is presented.
{"title":"A Refined Global Poincaré–Bendixson Annulus with the Limit Cycle of the Rayleigh System","authors":"Y. Li, A. A. Grin, A. V. Kuzmich","doi":"10.1134/s0012266124060028","DOIUrl":"https://doi.org/10.1134/s0012266124060028","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> New methods for constructing two Dulac–Cherkas functions are developed using which a\u0000better, depending on the parameter <span>(lambda >0)</span>, inner\u0000boundary of the Poincaré–Bendixson annulus <span>(A(lambda ) )</span> is found for the Rayleigh system. A procedure is\u0000proposed for directly finding a polynomial whose zero level set contains a transversal oval used as\u0000the outer boundary of <span>(A(lambda ))</span>. An\u0000interval for <span>(lambda )</span> is specified with which the best outer boundary of\u0000the annulus <span>( A(lambda ))</span> is a closed contour composed of two\u0000arcs of the constructed oval and two arcs of unclosed curves of the zero level set of one of the\u0000Dulac–Cherkas functions. Thus, a refined global Poincaré–Bendixson annulus for the\u0000limit cycle of the Rayleigh system is presented.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252267","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 : 2024-09-19DOI: 10.1134/s0012266124060016
M. M. Vas’kovskii, P. P. Stryuk
Abstract
We study the unique solvability of the Cauchy problem for a mixed-type stochastic differential equation driven by the standard Brownian motion and fractional Brownian motions with Hurst exponents (H>1/4). We prove a theorem on the existence and uniqueness of strong solutions of these stochastic differential equations.
{"title":"Existence and Uniqueness of Strong Solutions of Mixed-Type Stochastic Differential Equations Driven by Fractional Brownian Motions with Hurst Exponents $$H>1/4 $$","authors":"M. M. Vas’kovskii, P. P. Stryuk","doi":"10.1134/s0012266124060016","DOIUrl":"https://doi.org/10.1134/s0012266124060016","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the unique solvability of the Cauchy problem for a mixed-type stochastic\u0000differential equation driven by the standard Brownian motion and fractional Brownian motions\u0000with Hurst exponents <span>(H>1/4)</span>. We prove a\u0000theorem on the existence and uniqueness of strong solutions of these stochastic differential\u0000equations.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252266","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 : 2024-09-19DOI: 10.1134/s0012266124060041
L. M. Kozhevnikova
Abstract
The paper considers a second-order quasilinear elliptic equation with an integrable right-hand side. Restrictions on the structure of the equation are stated in terms of the generalized (N )-function. Unlike the author’s previous papers, there is no sign condition on the lower-order term of the equation. The existence of a renormalized solution of the Dirichlet problem for this equation is proved in nonreflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain.
{"title":"Existence of a Renormalized Solution of a Quasilinear Elliptic Equation without the Sign Condition on the Lower-Order Term","authors":"L. M. Kozhevnikova","doi":"10.1134/s0012266124060041","DOIUrl":"https://doi.org/10.1134/s0012266124060041","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper considers a second-order quasilinear elliptic equation with an integrable\u0000right-hand side. Restrictions on the structure of the equation are stated in terms of the\u0000generalized <span>(N )</span>-function. Unlike the author’s previous papers,\u0000there is no sign condition on the lower-order term of the equation. The existence of a renormalized\u0000solution of the Dirichlet problem for this equation is proved in nonreflexive\u0000Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268536","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 : 2024-09-19DOI: 10.1134/s001226612406003x
A. V. Aksenov, A. D. Polyanin
Abstract
We study the Monge–Ampère equation with three independent variables, which occurs in electron magnetohydrodynamics. A group analysis of this strongly nonlinear partial differential equation is carried out. An eleven-parameter transformation preserving the form of the equation is found. A formula is obtained that permits one to construct multiparameter families of solutions based on simpler solutions. Two-dimensional reductions leading to simpler partial differential equations with two independent variables are considered. One-dimensional reductions are described that permit one to obtain self-similar and other invariant solutions that satisfy ordinary differential equations. Exact solutions with additive, multiplicative, and generalized separation of variables are constructed, many of which admit representation in elementary functions. The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial–boundary value problems described by strongly nonlinear partial differential equations.
{"title":"Group Analysis, Reductions, and Exact Solutions of the Monge–Ampère Equation in Magnetic Hydrodynamics","authors":"A. V. Aksenov, A. D. Polyanin","doi":"10.1134/s001226612406003x","DOIUrl":"https://doi.org/10.1134/s001226612406003x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the Monge–Ampère equation with three independent variables, which\u0000occurs in electron magnetohydrodynamics. A group analysis of this strongly nonlinear partial\u0000differential equation is carried out. An eleven-parameter transformation preserving the form of the\u0000equation is found. A formula is obtained that permits one to construct multiparameter families of\u0000solutions based on simpler solutions. Two-dimensional reductions leading to simpler partial\u0000differential equations with two independent variables are considered. One-dimensional reductions\u0000are described that permit one to obtain self-similar and other invariant solutions that satisfy\u0000ordinary differential equations. Exact solutions with additive, multiplicative, and generalized\u0000separation of variables are constructed, many of which admit representation in elementary\u0000functions. The obtained results and exact solutions can be used to evaluate the accuracy and\u0000analyze the adequacy of numerical methods for solving initial–boundary value problems described\u0000by strongly nonlinear partial differential equations.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252268","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 : 2024-09-19DOI: 10.1134/s0012266124060089
P. P. Matus
Abstract
The article develops the theory of stability of linear operator schemes for operator inequalities and nonlinear nonstationary initial–boundary value problems of mathematical physics with nonlinearities of unbounded growth. Based on sufficient conditions for the stability of A.A. Samarskii’s two- and three-level difference schemes, the corresponding a priori estimates for operator inequalities are obtained under the condition of the criticality of the difference schemes under consideration, i.e., when the difference solution and its first time derivative are nonnegative at all nodes of the grid domain. The results obtained are applied to the analysis of the stability of difference schemes that approximate the Fisher and Klein–Gordon equations with nonlinear right-hand sides.
{"title":"Using Operator Inequalities in Studying the Stability of Difference Schemes for Nonlinear Boundary Value Problems with Nonlinearities of Unbounded Growth","authors":"P. P. Matus","doi":"10.1134/s0012266124060089","DOIUrl":"https://doi.org/10.1134/s0012266124060089","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The article develops the theory of stability of linear operator schemes for operator\u0000inequalities and nonlinear nonstationary initial–boundary value problems of mathematical physics\u0000with nonlinearities of unbounded growth. Based on sufficient conditions for the stability of\u0000A.A. Samarskii’s two- and three-level difference schemes, the corresponding a priori estimates for\u0000operator inequalities are obtained under the condition of the criticality of the difference schemes\u0000under consideration, i.e., when the difference solution and its first time derivative are nonnegative\u0000at all nodes of the grid domain. The results obtained are applied to the analysis of the stability of\u0000difference schemes that approximate the Fisher and Klein–Gordon equations with nonlinear\u0000right-hand sides.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252304","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 : 2024-09-19DOI: 10.1134/s0012266124060065
S. P. Zubova, E. V. Raetskaya
Abstract
A method for constructing a feedback matrix to solve the spectrum allocation (spectrum control; pole assignment) problem for a linear dynamical system is given. A new proof of the well-known theorem about the connection between the complete controllability of a dynamical system and the existence of a feedback matrix is formed in the process of constructing the cascade decomposition method. The entire set of arbitrary elements affecting the nonuniqueness of the matrix is identified. Examples of constructing a feedback matrix in the case of a real spectrum and in the presence of complex conjugate eigenvalues as well as for the case of multiple eigenvalues are given. The stability of the specified spectrum under small perturbations of system parameters with a fixed feedback matrix is studied.
{"title":"Solution of the Spectrum Allocation Problem for a Linear Control System with Closed Feedback","authors":"S. P. Zubova, E. V. Raetskaya","doi":"10.1134/s0012266124060065","DOIUrl":"https://doi.org/10.1134/s0012266124060065","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A method for constructing a feedback matrix to solve the spectrum allocation (spectrum\u0000control; pole assignment) problem for a linear dynamical system is given. A new proof of the\u0000well-known theorem about the connection between the complete controllability of a dynamical\u0000system and the existence of a feedback matrix is formed in the process of constructing the cascade\u0000decomposition method. The entire set of arbitrary elements affecting the nonuniqueness of the\u0000matrix is identified. Examples of constructing a feedback matrix in the case of a real spectrum\u0000and in the presence of complex conjugate eigenvalues as well as for the case of multiple eigenvalues\u0000are given. The stability of the specified spectrum under small perturbations of system parameters\u0000with a fixed feedback matrix is studied.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252271","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 : 2024-09-19DOI: 10.1134/s0012266124060053
V. G. Zamuraev
Abstract
The paper considers an optimal control problem in which the plant is described by a linear functional equation in a Hilbert space and a control action is a change of the space. Sufficient conditions for the existence of a solution are obtained. The results are generalized to the case in which the plant is described by a linear variational inequality.
{"title":"Existence of Optimal Sets for Linear Variational Equations and Inequalities","authors":"V. G. Zamuraev","doi":"10.1134/s0012266124060053","DOIUrl":"https://doi.org/10.1134/s0012266124060053","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper considers an optimal control problem in which the plant is described by a linear\u0000functional equation in a Hilbert space and a control action is a change of the space. Sufficient\u0000conditions for the existence of a solution are obtained. The results are generalized to the case in\u0000which the plant is described by a linear variational inequality.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252270","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 : 2024-09-11DOI: 10.1134/s0012266124050069
R. V. Brizitskii, N. N. Maksimova
Abstract
We study a two-parameter multiplicative control problem for a model of electron-induced charging of an inhomogeneous polar dielectric. Sharp local stability estimates for its optimal solutions with respect to small perturbations of both the cost functionals and the given function of the boundary value problem are derived. For one of the controls, the relay property or the bang–bang principle is established.
{"title":"Multiplicative Control Problems for the Diffusion–Drift Charging Model of an Inhomogeneous Polar Dielectric","authors":"R. V. Brizitskii, N. N. Maksimova","doi":"10.1134/s0012266124050069","DOIUrl":"https://doi.org/10.1134/s0012266124050069","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study a two-parameter multiplicative control problem for a model of electron-induced\u0000charging of an inhomogeneous polar dielectric. Sharp local stability estimates for its optimal\u0000solutions with respect to small perturbations of both the cost functionals and the given function of\u0000the boundary value problem are derived. For one of the controls, the relay property or the\u0000bang–bang principle is established.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175365","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 : 2024-09-11DOI: 10.1134/s001226612405001x
A. Yu. Aleksandrov
Abstract
A coupled system describing the interaction of a differential subsystem with nonlinearities of a sector type and a linear difference subsystem is considered. It is assumed that the system is positive. A diagonal Lyapunov–Krasovskii functional is constructed, and conditions are determined under which the absolute stability of the system can be proved with the use of such a functional. In the case of power-law nonlinearities, estimates for the rate of convergence of the solution to the origin are obtained. The stability of the corresponding system with parameter switching is analyzed. Sufficient conditions guaranteeing the asymptotic stability of the zero solution for any admissible switching law are obtained.
{"title":"Construction of Diagonal Lyapunov–Krasovskii Functionals for a Class of Positive Differential-Algebraic Systems","authors":"A. Yu. Aleksandrov","doi":"10.1134/s001226612405001x","DOIUrl":"https://doi.org/10.1134/s001226612405001x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A coupled system describing the interaction of a differential subsystem with nonlinearities\u0000of a sector type and a linear difference subsystem is considered. It is assumed that the system is\u0000positive. A diagonal Lyapunov–Krasovskii functional is constructed, and conditions are\u0000determined under which the absolute stability of the system can be proved with the use of such a\u0000functional. In the case of power-law nonlinearities, estimates for the rate of convergence of the\u0000solution to the origin are obtained. The stability of the corresponding system with parameter\u0000switching is analyzed. Sufficient conditions guaranteeing the asymptotic stability of the zero\u0000solution for any admissible switching law are obtained.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175335","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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