Том 219, № 2 (2016)
- Год: 2016
- Статей: 14
- URL: https://journals.rcsi.science/1072-3374/issue/view/14786
Article
Spatially Inhomogeneous Solutions for a Modified Kuramoto–Sivashinsky Equation
Аннотация
We study the periodic boundary value problem for a modified Kuramoto–Sivashinsky equation which can serve as a mathematical model describing formation of nanorelief on the surface of a planar target under the action of ion flux. We show that, as in the case of the traditional Kuramoto–Sivashinsky equation, it is possible to obtain spatially inhomogeneous solutions under the condition that the homogeneous equilibrium states can change the stability. We consider local bifurcations. We find sufficient conditions for the existence of shortwave solutions. Bibliography: 11 titles.
Accompanying Distributions of Singular Differential Operators
Аннотация
To study properties of solutions to differential equations by using the Fourier transform, we generalize the notion of accompanying distribution of a differential operator to the case of the Fourier-Bessel transform. Bibliography: 2 titles.
The Study of Discrete Mappings in TQ-Space. Basic principles
Аннотация
For multidimensional discrete mappings of the form sk+1 = g(sk, p) we develop the method of symbolic CTQ-analysis and study the main properties of the T-alphabet. We formulate basic principles for the analytic study of discrete mappings in terms of the proposed formalism. Bibliography: 15 titles. Illustrations: 5 figures.
Singular Space-Time Transformations. Towards One Method For Solving the Painlevé Problem
Аннотация
We develop the method of singular space-time transformations for systems with impact and friction. We consider a mathematical model of a Painlevé problem concerning impacts of an absolutely rigid bar with a rough surface. Using the method of singular space-time change of variables, we obtain the model of an absolutely rigid body by passing to the limit as the generalized rigidity parameter tends to infinity.
A Criterion for the Existence of Nontrivial Solutions to the Homogeneous Schwarz Problem
Аннотация
We consider the homogeneous Schwarz problem for Douglis analytic functions with twodimensional matrices possessing multiple eigenvalues. We find a necessary and sufficient condition for the existence of a nontrivial solution representable as a quadratic vectorvalued form.
Estimates of Homogenization for the Beltrami Equation
Аннотация
We study the homogenization of the Riemann–Hilbert boundary value problem for the Beltrami equation with an oscillating ε-periodic coefficient, where ε > 0 is a small parameter. The homogenized problem has a similar form, but with a constant coefficient. We prove error estimates of homogenization in the L2- and W21 -norms of order \( O\left(\sqrt{\varepsilon}\right) \). Bibliography: 9 titles.
Optimal Control for Quasilinear Degenerate Distributed Systems of Higher Order
Аннотация
We consider the problem for a distributed control with compromise quality functional for systems whose states are described by evolution equations that are unsolved with respect to the higher order time derivative. We establish the solvability of the problem for linear and quasilinear equations. The results are illustrated by an example. Bibliography: 12 titles.
A Problem for a Pseudohyperbolic Equation with Nonlocal Boundary Condition
Аннотация
We prove the existence and uniqueness of a weak solution to the initial-boundary value problem for a fourth order pseudohyperbolic equation in a cylinder with nonlocal boundary conditions containing the first and second order time-derivatives amd the integral of the sought solution. Bibliography: 3 titles.
Periodic Solutions to the Wave Equation with Homogeneous Boundary Conditions
Аннотация
We study time-periodic solutions to a quasilinear wave equation with homogeneous boundary conditions. We prove the existence of countably many periodic solutions in the case of boundary conditions of the third kind provided that the nonlinear term has power growth. It is shown that the Lp-norms of periodic solutions can be as large as desired. If the nonlinear term satisfies the nonresonance condition at infinity, we establish the existence of at least one periodic solution. We formulate a condition for the uniqueness of a periodic solution. Bibliography: 15 titles.
Random Perturbations of Autoresonance in Oscillating Systems with Small Dissipation
Аннотация
We consider the system of differential equations describing the initial step of capture of nonlinear oscillations in autoresonance under weak dissipation. We study the stability in probability of resonance solutions with unboundedly growing amplitude under persistent random perturbations. Bibliography: 11 titles. Illustrations: 1 Figure
Stabilization of Solutions to the Dirichlet Problem in a Cylindrical Domain for the Parabolic p-Laplacian
Аннотация
We prove a criterion for pointwise stabilization of bounded solutions to nonlinear parabolic p-Laplacian type equations in a cylindrical domain with an unbounded base. The criterion is formulated in terms of a Wiener series or integral and can be regarded as the condition of regularity of a point at infinity. Bibliography: 32 titles.
The Nikol’Skii Type Regularity of Solutions to Nonlinear Problems in Domains with Hölder Boundary
Аннотация
We study the regularity of solutions to problems of minimization of integral functionals, the Dirichlet and Neumann problems for elliptic operators of order 2m in domains with Hölder boundary on a compact Riemannian manifold. We establish interactions between the smoothness of the right-hand side, the regularity of the boundary, and the smoothness of solutions to the problems under consideration. Bibliography: 10 titles.
Integrable Systems with Variable Dissipation on the Tangent Bundle of a Sphere
Аннотация
Many problems of multidimensional dynamics involve systems for which the spaces of states are spheres of finite dimension and the spaces of phases are the tangent bundles of such spheres. We study conservative systems and present nonconservative force fields such that the systems involving such forces possess a complete collection of first integrals that are expressed through a finite combination of elementary functions and, in general, are transcendental functions of their variables. Bibliography: 32 titles.