Volume 236, Nº 6 (2019)
- Ano: 2019
- Artigos: 8
- URL: https://journals.rcsi.science/1072-3374/issue/view/14983
Article
Second Boundary-Value Problem for the Lavrent’ev–Bitsadze Equation in a Rectangular Domain with Two Degeneration Lines
Resumo
For a mixed-type equation, we examine the second boundary-value problem and by using the spectral method prove the uniqueness and existence of solutions. The uniqueness criterion is proved based on the completeness property of the biorthogonal system of functions corresponding to the onedimensional spectral problem. A solution of the problem is constructed as the sum of a biorthogonal series.
Special Class of Liouville-Type Hyperbolic Equations
Resumo
In this paper, we obtain differential substitutions for a special class of Liouville-type equations that simplify these equations. We present necessary and sufficient conditions of the existence of y-integrals of the second order. We examine the principal case of necessary and sufficient conditions of the existence of y-integrals.
Initial-Boundary-Value Problem for Inhomogeneous Degenerate Equations of Mixed Parabolic-Hyperbolic Type
Resumo
We consider initial-boundary-value problems for three classes of inhomogeneous degenerate equations of mixed parabolic-hyperbolic type: mixed-type equations with degenerate hyperbolic part, mixed-type equations with degenerate parabolic part, and mixed-type equations with power degeneration. In each case, we state a criterion of uniqueness of a solution to the problem. We construct solutions as series with respect to the system of eigenfunctions of the corresponding one-dimensional spectral problem. We prove that the uniqueness of the solution and the convergence of the series depend on the ratio of sides of the rectangular from the hyperbolic part of the mixed domain. In the proof of the existence of solutions to the problem, small denominators appear that impair the convergence of series constructed. In this connection, we obtain estimates of small denominators separated from zero and the corresponding asymptotics, which allows us, under certain conditions, to prove that the solution constructed belongs to the class of regular solutions.
Noncommutative Geometry and Analysis
Resumo
One of the main problems of noncommutative geometry is the translation of fundamental notions of analysis, topology, and differential geometry onto the language of Banach algebras. In this paper, we present a number of results of this kind focusing the attention on the noncommutative interpretation of the notions of differential and integral. Our presentation is based on the monographs Noncommutative Geometry by A. Connes and Elements of Noncommutative Geometry by J. M. Gracia-Bondia, J. C. Varilly, and H. Figueroa.
On Analytical in a Sector Resolving Families of Operators for Strongly Degenerate Evolution Equations of Higher and Fractional Orders
Resumo
In this paper, we study a class of linear evolution equations of fractional order that are degenerate on the kernel of the operator under the sign of the derivative and on its relatively generalized eigenvectors. We prove that in the case considered, in contrast to the case of first-order degenerate equations and equations of fractional order with weak degeneration (i.e., degeneration only on the kernel of the operator under the sign of the derivative), the family of analytical in a sector operators does not vanish on relative generalized eigenspaces of the operator under the sign of the derivative, has a singularity at zero, and hence does not determine any solution of a strongly degenerate equation of fractional order. For the case of a strongly degenerate equation of integer order this fact does not hold, but the behavior of the family of resolving operators at zero cannot be examined by ordinary method.
Vortex Steady Planar Entropic Flows of Ideal Gases
Resumo
We find all solutions to the submodel of vortex, steady, planar, barotropic, entropic flows of an ideal gas and show that possible motions are exhausted by rectilinear motions under a constant pressure and motions along concentric circles. We present a group classification of the model of planar, vortex, entropic, nonbarotropic flows, examine invariant submodels, and propose a physical interpretation of certain solutions.
New Examples of Integrable Systems with Dissipation on the Tangent Bundles of Multidimensional Spheres
Resumo
In many problems of multidimensional dynamics, systems appear whose state spaces are spheres of finite dimension. Clearly, phase spaces of such systems are tangent bundles of these spheres. In this paper, we examine nonconservative force fields in the dynamics of a multidimensional rigid body in which the system possesses a complete set of first integrals that can be expressed as finite combinations of elementary transcendental functions. We consider the case where the moment of nonconservative forces depends on the tensor of angular velocity.
Modeling of the Inner Structure of Tropical Cyclones: Equation of Flow on Wind Trajectories
Resumo
We consider a system of equations that describes the air motion inside tropical cyclones, including consequences of the condensation of water vapor. We examine the system of nonlinear ordinary differential equations on wind trajectories in the plane of radial and vertical variables. We prove the unique solvability of the system of differential equations on a segment suitable for modeling of tropical cyclones.