Differential Equations最新文献

{"title":"On the Asymptotic Behavior of Solutions of Third-Order Binomial Differential Equations","authors":"Ya. T. Sultanaev, N. F. Valeev, E. A. Nazirova","doi":"10.1134/s0012266124020095","DOIUrl":"https://doi.org/10.1134/s0012266124020095","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper discusses the development of a method for constructing asymptotic formulas as\u0000<span>(xto infty )</span> for the fundamental solution system of two-term\u0000singular symmetric differential equations of odd order with coefficients in a broad class of\u0000functions that allow oscillation (with relaxed regularity conditions that do not satisfy the classical\u0000Titchmarsh–Levitan regularity conditions). Using the example of a third-order binomial equation\u0000<span>(({i}/{2})bigl [(p(x)y^{prime })^{prime prime }+(p(x)y^{prime prime })^{prime }bigr ] +q(x)y =lambda y)</span>, the asymptotics of solutions in\u0000the case of various behavior of the coefficients <span>(q(x))</span> and\u0000<span>(h(x)=-1+{1}big /{sqrt {p(x)}})</span> is studied. New asymptotic\u0000formulas are obtained for the case in which <span>(h(x) notin L_1[1,infty ) )</span>.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512205","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}

{"title":"Solution of Some Problems for the String Vibration Equation in a Half-Strip by Quadratures","authors":"O. M. Jokhadze, S. S. Kharibegashvili","doi":"10.1134/s0012266124020034","DOIUrl":"https://doi.org/10.1134/s0012266124020034","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For the inhomogeneous string vibration equation in a half-strip, we consider a problem\u0000periodic in the spatial variable and a mixed problem. The solutions of these problems in the form\u0000of finite sums are obtained by quadratures. When solving these problems, we use the\u0000characteristic rectangle identity, Riemann invariants, and the method of characteristics.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550809","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}

{"title":"Gradient in the Problem of Controlling Processes Described by Linear Pseudohyperbolic Equations","authors":"A. M. Romanenkov","doi":"10.1134/s001226612402006x","DOIUrl":"https://doi.org/10.1134/s001226612402006x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper considers the problem of controlling processes whose mathematical model is an initial–boundary value problem for a pseudohyperbolic linear differential equation of high order in the spatial variable and second order in the time variable. The pseudohyperbolic equation is a generalization of the ordinary hyperbolic equation typical in vibration theory. As examples, we consider models of vibrations of moving elastic materials. For the model problems, an energy identity is established and conditions for the uniqueness of a solution are formulated. As an optimization problem, we consider the problem of controlling the right-hand side so as to minimize a quadratic integral functional that evaluates the proximity of the solution to the objective function. From the original functional, a transition is made to a majorant functional, for which the corresponding upper bound is established. An explicit expression for the gradient of this functional is obtained, and adjoint initial–boundary value problems are derived.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553245","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}

{"title":"Logistic Equation with Long Delay Feedback","authors":"S. A. Kashchenko","doi":"10.1134/s0012266124020010","DOIUrl":"https://doi.org/10.1134/s0012266124020010","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the local dynamics of the delay logistic equation with an additional feedback\u0000containing a large delay. Critical cases in the problem of stability of the zero equilibrium state are\u0000identified, and it is shown that they are infinite-dimensional. The well-known methods for\u0000studying local dynamics based on the theory of invariant integral manifolds and normal forms do\u0000not apply here. The methods of infinite-dimensional normalization proposed by the author are\u0000used and developed. As the main results, special nonlinear boundary value problems of parabolic\u0000type are constructed, which play the role of normal forms. They determine the leading terms of\u0000the asymptotic expansions of solutions of the original equation and are called quasinormal forms.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550811","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}

{"title":"Cascade Super-Twisting Observer for Linear Multiagent Systems without Communication","authors":"V. V. Fomichev, A. I. Samarin","doi":"10.1134/s0012266124020083","DOIUrl":"https://doi.org/10.1134/s0012266124020083","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper addresses the consensus problem (i.e., the agreement of state vectors) for a\u0000multiagent system consisting of identical linear agents. The study focuses on the case where there\u0000is no communication between agents, meaning there is no exchange of information, and agent\u0000control is achieved through the agents’ own sensors, providing incomplete information about the\u0000state vector of the agent and its neighbors, with the information possibly being noisy. To solve\u0000this problem, a linear protocol based on observer data for systems under uncertainty is proposed.\u0000Cascade observers based on the “super-twisting” method are suggested as such observers. Sufficient\u0000conditions are obtained for the existence of a controller where the observation error converges to\u0000zero under bounded disturbances. An example illustrating the proposed approach is provided.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550810","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}

{"title":"Solvability of an Initial–Boundary Value Problem for the Modified Kelvin–Voigt Model with Memory along Fluid Motion Trajectories","authors":"M. V. Turbin, A. S. Ustiuzhaninova","doi":"10.1134/s0012266124020046","DOIUrl":"https://doi.org/10.1134/s0012266124020046","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper deals with proving the weak solvability of an initial–boundary value problem for\u0000the modified Kelvin–Voigt model taking into account memory along the trajectories of motion of\u0000fluid particles. To this end, we consider an approximation problem whose solvability is established\u0000with the use of the Leray–Schauder fixed point theorem. Then, based on a priori estimates, we\u0000show that the sequence of solutions of the approximation problem has a subsequence that weakly\u0000converges to the solution of the original problem as the approximation parameter tends to zero.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552936","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}

{"title":"On the Solvability of Fredholm Boundary Integral Equations of the First Kind for the Three-Dimensional Transmission Problem on the Spectrum","authors":"A. A. Kashirin, S. I. Smagin","doi":"10.1134/s0012266124020058","DOIUrl":"https://doi.org/10.1134/s0012266124020058","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper considers two weakly singular Fredholm boundary integral equations of the first\u0000kind to each of which the three-dimensional Helmholtz transmission problem can be reduced. The\u0000properties of these equations are studied on the spectra, where they are ill posed. For the first\u0000equation, it is shown that its solution, if it exists on the spectrum, allows finding a solution of the\u0000transmission problem. The second equation in this case always has infinitely many solutions, with\u0000only one of them giving a solution of the transmission problem. The interpolation method for\u0000finding approximate solutions of the integral equations and the transmission problem in question\u0000is discussed.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553244","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}

{"title":"On the Spectrum of Nonself-Adjoint Dirac Operators with Two-Point Boundary Conditions","authors":"A. S. Makin","doi":"10.1134/s0012266124020022","DOIUrl":"https://doi.org/10.1134/s0012266124020022","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the spectral problem for the Dirac operator with arbitrary two-point boundary\u0000conditions and any square integrable potential <span>(V)</span>. Necessary and\u0000sufficient conditions for an entire function to be the characteristic determinant of such an operator\u0000are established. In the case of irregular boundary conditions, conditions are found under which a\u0000set of complex numbers is the spectrum of the problem under consideration.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550812","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}

{"title":"On Regularization of Classical Optimality Conditions in Convex Optimization Problems for Volterra-Type Systems with Operator Constraints","authors":"V. I. Sumin, M. I. Sumin","doi":"10.1134/s0012266124020071","DOIUrl":"https://doi.org/10.1134/s0012266124020071","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the regularization of classical optimality conditions—the Lagrange\u0000principle and the Pontryagin maximum principle—in a convex optimal control problem\u0000with an operator equality constraint and functional inequality constraints. The controlled system\u0000is specified by a linear functional–operator equation of the second kind of general form in the\u0000space <span>(L^m_2 )</span>, and the main operator on the right-hand side of\u0000the equation is assumed to be quasinilpotent. The objective functional of the problem is only\u0000convex (perhaps not strongly convex). Obtaining regularized classical optimality conditions is\u0000based on the dual regularization method. In this case, two regularization parameters are used, one\u0000of which is “responsible” for the regularization of the dual problem, and the other is contained in\u0000the strongly convex regularizing Tikhonov addition to the objective functional of the original\u0000problem, thereby ensuring the well-posedness of the problem of minimizing the Lagrange function.\u0000The main purpose of the regularized Lagrange principle and Pontryagin maximum principle is the\u0000stable generation of minimizing approximate solutions in the sense of J. Warga. The regularized\u0000classical optimality conditions\u0000</p><ol>\u0000<li>\u0000<span>1.</span>\u0000<p>Are formulated as existence theorems for minimizing approximate solutions in the original\u0000problem with a simultaneous constructive representation of these solutions. </p>\u0000</li>\u0000<li>\u0000<span>2.</span>\u0000<p>Are expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions.\u0000</p>\u0000</li>\u0000<li>\u0000<span>3.</span>\u0000<p>“Overcome” the properties of the ill-posedness of the classical optimality conditions and\u0000provide regularizing algorithms for solving optimization problems. </p>\u0000</li>\u0000</ol><p>Based on the perturbation method, an important property of the regularized\u0000classical optimality conditions obtained in the work is discussed in sufficient detail; namely, “in the\u0000limit” they lead to their classical counterparts. As an application of the general results obtained in\u0000the paper, a specific example of an optimal control problem associated with an integro-differential\u0000equation of the transport equation type is considered, a special case of which is a certain inverse\u0000final observation problem.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553246","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}

{"title":"Internal Transition Layer Structure in the Reaction–Diffusion Problem for the Case of a Balanced Reaction with a Weak Discontinuity","authors":"E. I. Nikulin, V. T. Volkov, D. A. Karmanov","doi":"10.1134/s0012266124010063","DOIUrl":"https://doi.org/10.1134/s0012266124010063","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For a singularly perturbed reaction–diffusion equation, we study the structure of the\u0000internal transition layer in the case of a balanced reaction with a weak discontinuity. The\u0000existence of solutions with an internal transition layer (contrast structures) is proved, the question\u0000of their stability is investigated, and asymptotic approximations to solutions of this type are\u0000obtained. It is shown that in the case of reaction balance, the presence of even a weak\u0000(asymptotically small) reaction discontinuity can lead to the formation of contrast structures of\u0000finite size, both stable and unstable.\u0000</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574631","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}