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Том 235, № 3 (2018)
- Год: 2018
- Статей: 8
- URL: https://bakhtiniada.ru/1072-3374/issue/view/14972
Article
Solvability of Some Integro-Differential Equations with Anomalous Diffusion in Two Dimensions
Аннотация
We study the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in two dimensions. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.
243-255
Application of the p-Adic Topology on ℤ to the Problem of Finding Solutions in Integers of an Implicit Linear Difference Equation
Аннотация
We study solutions in integers of an implicit linear inhomogeneous first order difference equation bxn+1 = axn + fn. Based on the p-adic topology on the ring of integers, we obtain a criterion for the existence of solutions and show that for a = 1 a typical (in the natural topological sense) equation has no integer solutions.
256-261
Smoothness of Functions in Spaces of the Finite Element Method
Аннотация
We find necessary and sufficient conditions for the generalized smoothness of the coordinate functions obtained from the approximate relations. We show that for the coordinate functions the smoothness on their supports is equivalent to that on the boundaries of supports. We obtain conditions for the continuity of Courant type finite element approximations and conditions for the uniqueness of a linear space of such approximations.
262-274
Linear Oscillations of Thin Plates with Corners and Cracks
Аннотация
We study linear oscillations of a thin plate in a bounded domain G ⊂ ℝ2 with finitely many corner points at the boundary ∂G. The time t runs the real axis. The boundary conditions are related to the rigidly fixed or free boundary of the plate. We describe the asymptotics of a solution to the problem near the corner points.
275-311
On Consequences of the Strong Convergence in Lebesgue–Orlich Spaces
Аннотация
We study the continuity in the sense of the strong topology for the flux function υ → l(υ) = |υ|p(⋅) − 2υ acting from the Lebesgue–Orlicz space Lp(⋅)(Ω, ℝm) to the dual Lp ′ (⋅)(Ω, ℝm), where p′(⋅) is the Hölder-conjugate exponent, under the assumption that p(·) is an L∞(Ω)-function such that 1 < α ≤ p(·) ≤ β < ∞. We obtain estimates for the convergence \( {\left\Vert l\left({\upsilon}_n\right)-l\left(\upsilon \right)\right\Vert}_{p^{\prime}\left(\cdot \right)}\to 0 \) with respect to the smallness order as ‖υn − υ‖p(⋅) → 0. The strong continuity of the energy functional \( \underset{\varOmega }{\int }{\left|\upsilon \right|}^{p\left(\cdot \right)} dx \) is a consequence of the strong continuity of the flux function.
312-321
322-333
Integrable Dynamic Systems with Dissipation and Finitely Many Degrees of Freedom
Аннотация
We establish the integrability for some classes of dynamic systems on the tangent bundles of multidimensional manifolds. We consider the case where the force fields possess variable dissipation. An example of a four-dimensional manifold is discussed in detail.
334-359
The Neumann Problem for the Generalized Hénon Equation
Аннотация
We study the behavior of radial solutions to the boundary value problem
in the unit ball B and prove the existence of nonradial positive solutions for some values of parameters. We obtain multiplicity results which are new even in the case p = 2.
360-373