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Authors = Dana Smetanová

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7 pages, 235 KiB  
Article
Nonlocal Problem for a Third-Order Equation with Multiple Characteristics with General Boundary Conditions
by Abdukomil Risbekovich Khashimov and Dana Smetanová
Axioms 2021, 10(2), 110; https://doi.org/10.3390/axioms10020110 - 2 Jun 2021
Cited by 1 | Viewed by 2332
Abstract
The article considers third-order equations with multiple characteristics with general boundary value conditions and non-local initial data. A regular solution to the problem with known methods is constructed here. The uniqueness of the solution to the problem is proved by the method of [...] Read more.
The article considers third-order equations with multiple characteristics with general boundary value conditions and non-local initial data. A regular solution to the problem with known methods is constructed here. The uniqueness of the solution to the problem is proved by the method of energy integrals. This uses the theory of non-negative quadratic forms. The existence of a solution to the problem is proved by reducing the problem to Fredholm integral equations of the second kind. In this case, the method of Green’s function and potential is used. Full article
(This article belongs to the Special Issue Boundary-Value and Spectral Problems)
8 pages, 226 KiB  
Article
On the Uniqueness Classes of Solutions of Boundary Value Problems for Third-Order Equations of the Pseudo-Elliptic Type
by Abdukomil Risbekovich Khashimov and Dana Smetanová
Axioms 2020, 9(3), 80; https://doi.org/10.3390/axioms9030080 - 16 Jul 2020
Cited by 1 | Viewed by 2030
Abstract
The paper is devoted to solutions of the third order pseudo-elliptic type equations. An energy estimates for solutions of the equations considering transformation’s character of the body form were established by using of an analog of the Saint-Venant principle. In consequence of this [...] Read more.
The paper is devoted to solutions of the third order pseudo-elliptic type equations. An energy estimates for solutions of the equations considering transformation’s character of the body form were established by using of an analog of the Saint-Venant principle. In consequence of this estimate, the uniqueness theorems were obtained for solutions of the first boundary value problem for third order equations in unlimited domains. The energy estimates are illustrated on two examples. Full article
(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)
16 pages, 304 KiB  
Article
Infinitesimal Transformations of Locally Conformal Kähler Manifolds
by Yevhen Cherevko, Volodymyr Berezovski, Irena Hinterleitner and Dana Smetanová
Mathematics 2019, 7(8), 658; https://doi.org/10.3390/math7080658 - 24 Jul 2019
Cited by 1 | Viewed by 3063
Abstract
The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also [...] Read more.
The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. We have also calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding Kählerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to some subgroup of the homothetic group of the coresponding local Kählerian metric. Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
15 pages, 247 KiB  
Article
Higher Order Hamiltonian Systems with Generalized Legendre Transformation
by Dana Smetanová
Mathematics 2018, 6(9), 163; https://doi.org/10.3390/math6090163 - 10 Sep 2018
Cited by 3 | Viewed by 3239
Abstract
The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton [...] Read more.
The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton equations and the Euler–Lagrange equations are studied. The theory is illustrated on examples of Hamiltonian systems satisfying the following conditions: (a) the Hamiltonian system is strongly regular and the Legendre transformation exists; (b) the Hamiltonian system is strongly regular and the Legendre transformation does not exist; (c) the Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition. Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
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